472A Chapter 9 Factoring Pacing suggestions for the entire year can be found on pages T20–T21. Factoring Chapter Overview and Pacing Factoring Chapter Overview and Pacing PACING (days) Regular Block Basic/ Basic/ Average Advanced Average Advanced Factors and Greatest Common Factors (pp. 474–479) 1 1 0.5 0.5 • Find prime factorizations of integers and monomials. • Find the greatest common factors of integers and monomials. Factoring Using the Distributive Property (pp. 480–486) 2 1 1 0.5 Preview: Use algebra tiles and a product mat to factor binomials. (with 9-2 • Factor polynomials by using the Distributive Property. Preview) • Solve quadratic equations of the form ax 2 bx 0. Factoring Trinomials: x 2 bx c (pp. 487–494) 3 2 2 1 Preview: Use algebra tiles to factor trinomials. (with 9-3 (with 9-3 • Factor trinomials of the form x 2 bx c. Preview) Preview) • Solve equations of the form x 2 bx c 0. Factoring Trinomials: ax 2 bx c (pp. 495–500) 2 2 1 1 • Factor trinomials of the form ax 2 bx c. • Solve equations of the form ax 2 bx c 0. Factoring Differences of Squares (pp. 501–506) 2 2 1 1 • Factor binomials that are the differences of squares. • Solve equations involving the differences of squares. Perfect Squares and Factoring (pp. 508–514) 2 2 1 1 • Factor perfect square trinomials. • Solve equations involving perfect squares. Study Guide and Practice Test (pp. 515–519) 1 1 1 0.5 Standardized Test Practice (pp. 520–521) Chapter Assessment 1 1 0.5 0.5 TOTAL 14 12 8 6 LESSON OBJECTIVES
Transcript
472A Chapter 9 Factoring
Pacing suggestions for the entire year can be found on pages T20–T21.
FactoringChapter Overview and PacingFactoringChapter Overview and Pacing
PACING (days)Regular Block
Basic/ Basic/ Average Advanced Average Advanced
Factors and Greatest Common Factors (pp. 474–479) 1 1 0.5 0.5• Find prime factorizations of integers and monomials.• Find the greatest common factors of integers and monomials.
Factoring Using the Distributive Property (pp. 480–486) 2 1 1 0.5Preview: Use algebra tiles and a product mat to factor binomials. (with 9-2• Factor polynomials by using the Distributive Property. Preview)• Solve quadratic equations of the form ax2 � bx � 0.
Factoring Trinomials: x2 � bx � c (pp. 487–494) 3 2 2 1Preview: Use algebra tiles to factor trinomials. (with 9-3 (with 9-3• Factor trinomials of the form x2 � bx � c. Preview) Preview)• Solve equations of the form x2 � bx � c � 0.
Factoring Trinomials: ax2 � bx � c (pp. 495–500) 2 2 1 1• Factor trinomials of the form ax2 � bx � c.• Solve equations of the form ax2 � bx � c � 0.
Factoring Differences of Squares (pp. 501–506) 2 2 1 1• Factor binomials that are the differences of squares.• Solve equations involving the differences of squares.
Mathematical Connections and BackgroundMathematical Connections and Background
Factors and Greatest Common FactorsThe factors of a given number are all the numbers
that divide the number evenly. This includes the numberitself and 1. The factors of a number can be found bydetermining all the pairs of numbers whose product isthat number. Natural numbers greater than 1 are classi-fied as either prime or composite. Prime numbers haveexactly two factors while composite numbers have morethan two factors. The number 1 is neither prime nor com-posite. A prime factorization is the expression of a num-ber as the product of factors that are all prime numbers.The prime factorization of a negative integer is expressedas the product �1 and prime numbers. Monomials can bewritten in factored form. A monomial in factored form isthe product of prime numbers and variables. Variables,however, cannot have an exponent greater than 1. So x3
must be written as x � x � x in factored form. Prime factorizations are used to determine the
greatest common factor (GCF) of two or more integers ormonomials. The GCF is the product of all the commonprime factors of the two integers or monomials. If 1 isthe only common factor, then they are relatively prime.
Factoring Using the Distributive PropertyMany polynomials also have factors. Some poly-
nomials are the product of a polynomial and a monomial.Reverse the process of multiplying a polynomial by amonomial to factor using the Distributive Property. Firstfind the greatest common factor of the terms of the poly-nomial. If the GCF is not 1, then rewrite each term as theproduct of the GCF and its remaining factors. Then usethe Distributive Property to factor out the GCF.
If a polynomial has four or more terms you can fac-tor by grouping. Group terms in pairs that have commonfactors. Use the Distributive Property to factor the GCFfrom each pair of terms. The binomials in each pair offactored terms should be identical. Use the DistributiveProperty to factor out the common binomial factor. Theremaining factors are grouped to form a second binomial.It may appear that the factored pairs do not have identi-cal binomials, but one may be the additive inverse of theother. Write one as the product of �1 and its additiveinverse. Then multiply the GCF of that pair by �1.
If the product of two factors is 0, then at least oneof them is 0 according to the Zero Product Property. If anequation has the form ab � 0 or can be written in thisform by factoring, then the Zero Product Property can beapplied to solve the equation. Set each factor equal to 0and solve each resulting equation.
Prior KnowledgePrior KnowledgeStudents studied prime numbers and great-est common factors in previous courses.They also found the prime factorization ofnumbers. In Chapter 8, students learned therules for dividing monomials.
This Chapter
Future ConnectionsFuture ConnectionsFactoring polynomials is used to solve manyreal-world problems. It is basic to studyingmore about polynomial equations and functions.
Continuity of InstructionContinuity of Instruction
This ChapterThis chapter covers the factoring of inte-gers, monomials and polynomials. Studentslearn to find the greatest common factor ofthe monomials in a polynomial. They usethe GCF and the Distributive Property tofactor polynomials. Students also apply theZero Product Property to solve quadratic
equations.
Chapter 9 Factoring 472D
Factoring Trinomials: x2 � bx � cThe FOIL method was used to multiply two
binomials. Reverse the FOIL method to factor a quad-ratic polynomial of the form x2 � bx � c into two bi-nomials. Find two numbers m and n whose product is cand whose sum is b. The two numbers are the lastterms of the two binomials (x � m) and (x � n).
If b is negative and c is positive then m and nmust both be negative. If c is negative, then m and nmust have different signs. This is because the productof two numbers with different signs is negative.
The Zero Product Property can be used tosolve some quadratic equations written in the form x2 � bx � c � 0. Factor the trinomial, and then seteach factor equal to 0. Solve each equation to find thesolution of the quadratic equation. Be sure to checkthe solutions in the original equation.
Factoring Trinomials: ax2 � bx � cTo factor a trinomial in which the coefficient of
x2 is not 1, first check to see if the terms of the polyno-mial have a GCF. If so, factor it out. If the coefficient ofx2 is still not 1, or there is no GCF, factor ax2 � bx � cby making an organized list of the factors of theproduct of a and c. For example, to factor 8x2 � 5x � 3,make an organized list of the factors of 8 � (�3), or �24.Look for a pair of factors, m and n, whose sum isequal to b in the trinomial. Then rewrite the trinomial,replacing bx with mx � nx. The new polynomial hasfour terms. Use the factoring by grouping techniqueshown in Lesson 9-2 to factor this polynomial intotwo binomial factors. A polynomial that cannot befactored is a prime polynomial. Use the above methodand the Zero Product Property to solve equations inthe form ax2 � bx � c � 0.
Factoring Differences of SquaresLesson 8-8 discussed the pattern for the prod-
uct of a sum and a difference: (a � b)(a � b) � a2 � b2.The binomial a2 � b2 is called the difference of twosquares and can be factored as the product of a sumand a difference: a2 � b2 � (a � b)(a � b). To do this,identify a and b, the square roots of the first and lastterms, respectively. Then apply the pattern.
Other aspects of factoring to watch for are fac-toring out a common factor, applying a technique morethan once, and applying several techniques. Use anyof the appropriate techniques and the Zero ProductProperty to solve many polynomial equations.
Perfect Squares and FactoringSome trinomials have patterns that make their
factoring easier. In Lesson 8-8, students learned aboutpatterns for the square of a sum, (a � b)2 � a2 � 2ab � b2,and the square of a difference, (a � b)2 � a2 � 2ab � b2.These products, a2 � 2ab � b2 and a2 � 2ab � b2, arecalled perfect square trinomials, because they are theresult of squaring a binomial. To recognize a perfectsquare trinomial, first determine if the first and lastterms are perfect squares. Then find the square roots ofthe first and last terms, checking to see if twice theproduct of these square roots is equal to the middleterm of the trinomial. If the trinomial is a perfectsquare, and the middle term is positive use the patterna2 � 2ab � b2 � (a � b)2 to factor it. If the middle termis negative, use the pattern a2 � 2ab � b2 � (a � b)2. Itis important to note that the last term of a perfectsquare trinomial cannot be negative.
If one side of an equation is a perfect square orcan be written as a perfect square, then the Square RootProperty can be applied to solve the equation. TheSquare Root Property allows you to take the squareroot of each side of an equation, so long as both thepositive and negative square roots of a number aretaken into account. So, for any number n that is greaterthan 0, if x2 � n, then x � ��n�. Two solutions resultfrom such equations, one using the positive squareroot and one using the negative square root.
Additional mathematical information and teaching notesare available in Glencoe’s Algebra 1 Key Concepts:Mathematical Background and Teaching Notes, which is available at www.algebra1.com/key_concepts. The lessons appropriate for this chapter are as follows.• Solving Equations by Factoring (Lesson 27)
TestCheck and Worksheet BuilderThis networkable software has three modules for interventionand assessment flexibility:• Worksheet Builder to make worksheet and tests• Student Module to take tests on screen (optional)• Management System to keep student records (optional)
Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests.
5-Minute Check TransparenciesPrerequisite Skills Workbook, pp. 13–14Quizzes, CRM pp. 573–574Mid-Chapter Test, CRM p. 575Study Guide and Intervention, CRM pp. 523–524,
529–530, 535–536, 541–542, 547–548, 553–554
MixedReview
Cumulative Review, CRM p. 576 pp. 479, 486, 494, 500, 506, 514
ErrorAnalysis
Find the Error, TWE pp. 492, 498, 504Unlocking Misconceptions, TWE p. 497Tips for New Teachers, TWE pp. 490, 509
Find the Error, pp. 492, 498, 504Common Misconceptions,
pp. 483, 502
StandardizedTest Practice
TWE pp. 520–521Standardized Test Practice, CRM pp. 577–578
Standardized Test Practice CD-ROM
www.algebra1.com/standardized_test
pp. 479, 486, 494, 500, 503,505, 514, 519, 520–521
Open-EndedAssessment
Modeling: TWE pp. 479, 506Speaking: TWE pp. 486, 494Writing: TWE pp. 500, 514Open-Ended Assessment, CRM p. 571
Writing in Math, pp. 479, 485,494, 500, 506, 514
Open Ended, pp. 477, 484, 492,498, 504, 512
Standardized Test, p. 521
ChapterAssessment
Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 559–564
Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 565–570
Vocabulary Test/Review, CRM p. 572
TestCheck and Worksheet Builder(see below)
MindJogger Videoquizzes www.algebra1.com/
vocabulary_reviewwww.algebra1.com/chapter_test
Study Guide, pp. 515–518Practice Test, p. 519
Additional Intervention ResourcesThe Princeton Review’s Cracking the SAT & PSATThe Princeton Review’s Cracking the ACTALEKS
and Assessmentand AssessmentA
SSES
SMEN
TIN
TER
VEN
TIO
N
Type Student Edition Teacher Resources Technology/Internet
ALEKS is an online mathematics learning system thatadapts assessment and tutoring to the student’s needs.Subscribe at www.k12aleks.com.
For more information on Reading and Writing inMathematics, see pp. T6–T7.
Intervention at HomeParent and Student Study Guide Parents and students may work together to reinforce theconcepts and skills of this chapter. (Workbook, pp. 68–74 or log on to www.algebra1.com/parent_student)
Intervention TechnologyAlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum.
Reading and Writingin Mathematics
Reading and Writingin Mathematics
Glencoe Algebra 1 provides numerous opportunities toincorporate reading and writing into the mathematics classroom.
Student Edition
• Foldables Study Organizer, p. 473• Concept Check questions require students to verbalize
and write about what they have learned in the lesson.(pp. 477, 484, 492, 498, 504, 512)
• Reading Mathematics, p. 507 • Writing in Math questions in every lesson, pp. 479, 485,
494, 500, 506, 514• Reading Study Tip, pp. 489, 511• WebQuest, p. 479
Teacher Wraparound Edition
• Foldables Study Organizer, pp. 473, 515• Study Notebook suggestions, pp. 477, 480, 484, 488,
492, 498, 504, 507, 512 • Modeling activities, pp. 479, 506• Speaking activities, pp. 486, 494• Writing activities, pp. 500, 514• Differentiated Instruction, (Verbal/Linguistic), p. 475• Resources, pp. 472, 475, 478, 485, 493, 499,
505, 507, 513, 515
Additional Resources
• Vocabulary Builder worksheets require students todefine and give examples for key vocabulary terms asthey progress through the chapter. (Chapter 9 ResourceMasters, pp. vii-viii)
• Reading to Learn Mathematics master for each lesson(Chapter 9 Resource Masters, pp. 527, 533, 539, 545,551, 559)
• Vocabulary PuzzleMaker software creates crossword,jumble, and word search puzzles using vocabulary liststhat you can customize.
• Teaching Mathematics with Foldables provides suggestions for promoting cognition and language.
• Reading and Writing in the Mathematics Classroom• WebQuest and Project Resources• Hot Words/Hot Topics Sections 1.2, 3.2, 6.2
ELL
For more information on Intervention andAssessment, see pp. T8–T11.
Log on for student study help.• For each lesson in the Student Edition, there are Extra
Examples and Self-Check Quizzes.www.algebra1.com/extra_exampleswww.algebra1.com/self_check_quiz
• For chapter review, there is vocabulary review, test practice, and standardized test practice.www.algebra1.com/vocabulary_reviewwww.algebra1.com/chapter_testwww.algebra1.com/standardized_test
Have students read over the listof objectives and make a list ofany words with which they arenot familiar.
Point out to students that this isonly one of many reasons whyeach objective is important.Others are provided in theintroduction to each lesson.
472 Chapter 9 Factoring
Factoring
• factored form (p. 475)• factoring by grouping (p. 482)• prime polynomial (p. 497)• difference of squares (p. 501)• perfect square trinomials (p. 508)
Key Vocabulary
The factoring of polynomials can be used to solve a variety ofreal-world problems and lays the foundation for the further study ofpolynomial equations. Factoring is used to solve problems involvingvertical motion. For example, the height h in feet of a dolphin that jumps out of the water traveling at 20 feet per second is modeled by apolynomial equation. Factoring can be used to determine how long thedolphin is in the air. You will learn how to solve polynomial equations in Lesson 9-2.
• Lesson 9-1 Find the prime factorizations ofintegers and monomials.
• Lesson 9-1 Find the greatest common factors(GCF) for sets of integers and monomials.
• Lessons 9-2 through 9-6 Factor polynomials.
• Lessons 9-2 through 9-6 Use the Zero ProductProperty to solve equations.
Vocabulary BuilderThe Key Vocabulary list introduces students to some of the main vocabulary termsincluded in this chapter. For a more thorough vocabulary list with pronunciations ofnew words, give students the Vocabulary Builder worksheets found on pages vii andviii of the Chapter 9 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they addthese sheets to their study notebooks for future reference when studying for theChapter 9 test.
ELL
This section provides a review ofthe basic concepts needed beforebeginning Chapter 9. Pagereferences are included foradditional student help.
Additional review is provided inthe Prerequisite Skills Workbook,pp. 13–14.
Prerequisite Skills in the GettingReady for the Next Lesson sectionat the end of each exercise setreview a skill needed in the nextlesson.
Chapter 9 Factoring 473
Make this Foldable to help you organize your notes on
factoring. Begin with a sheet of plain 8�12
�" by 11" paper.
Label eachtab asshown.
9-19-29-39-49-59-6
Factoring
Open.Cut short side along folds to make tabs.
Open. Fold lengthwise,
leaving a tab on the right.
12
"Fold in thirds and then in half along
the width.
Reading and Writing As you read and study the chapter, write notes and examplesfor each lesson under its tab.
Prerequisite Skills To be successful in this chapter, you’ll need to masterthese skills and be able to apply them in problem-solving situations. Reviewthese skills before beginning Chapter 9.
For Lessons 9-2 through 9-6 Distributive Property
Rewrite each expression using the Distributive Property. Then simplify.(For review, see Lesson 1-5.)
Find each square root. (For review, see Lesson 2-7.)
13. �121� 11 14. �0.0064� 0.08 15. ��2356�� �
56
� 16. ��988�� �
27
�
�2x2 � 4xy � 96y2
�18y2 � 30y3 � 6y4
Fold in Sixths
Cut
Fold Again
Label
Chapter 9 Factoring 473
For PrerequisiteLesson Skill
9-2 Distributive Property, p. 479
9-3 Multiplying Polynomials, p. 486
9-4 Factoring by Grouping, p. 494
9-5 Square Roots, p. 500
9-6 Special Products, p. 506
Organization of Data and Questioning Before beginning eachlesson, ask students to think of one question that comes to mindas they skim through the lesson. Write the question on the front ofthe corresponding lesson tab. As students read and work throughthe lesson, ask them to record the answer to their question underthe tab. Students can also use their Foldables to take notes, recordconcepts, define terms, and record other questions that ariseabout factoring.
TM
For more informationabout Foldables, seeTeaching Mathematicswith Foldables.
5-Minute CheckTransparency 9-1 Use as a
quiz or review of Chapter 8.
Mathematical Background notesare available for this lesson on p. 472C.
are prime numbersrelated to the search for
extraterrestrial life?Ask students:• What is a prime number?
A prime number is any wholenumber, greater than one, whoseonly factors are one and itself.
• Why might a radio signal fromspace composed of primenumbers be significant?Sample answer: A signal composedof only prime numbers would seemto signify that it was sent byintelligent beings.
PRIME FACTORIZATION Recall that when two or more numbers aremultiplied, each number is a factor of the product. Some numbers, like 18, can be expressed as the product of different pairs of whole numbers. This can be shown geometrically. Consider all of the possible rectangles with whole numberdimensions that have areas of 18 square units.
The number 18 has 6 factors, 1, 2, 3, 6, 9, and 18. Whole numbers greater than 1 canbe classified by their number of factors.
Vocabulary• prime number• composite number• prime factorization• factored form• greatest common factor
(GCF)
Factors and Greatest Common Factors
474 Chapter 9 Factoring
are prime numbers related to the search for extraterrestrial life?are prime numbers related to the search for extraterrestrial life?
Classify Numbers as Prime or CompositeFactor each number. Then classify each number as prime or composite.
a. 36
To find the factors of 36, list all pairs of whole numbers whose product is 36.
1 � 36 2 � 18 3 � 12 4 � 9 6 � 6
Therefore, the factors of 36, in increasing order, are 1, 2, 3, 4, 6, 9, 12, 18, and 36.Since 36 has more than two factors, it is a composite number.
Example 1Example 1
1 � 18
2 � 9 3 � 6
Prime and Composite NumbersWords Examples
A whole number, greater than 1, whose only factors are 1 and itself, is called a .
2, 3, 5, 7, 11, 13, 17, 19
A whole number, greater than 1, that has more than two factors is called a .
4, 6, 8, 9, 10, 12, 14, 15, 16, 18
0 and 1 are neither prime nor composite.
composite number
prime number
Study TipListing FactorsNotice that in Example 1,6 is listed as a factor of 36only once.
• Find prime factorizations of integers and monomials.
• Find the greatest common factors of integers and monomials.
In the search for extraterrestrial life, scientists listen to radio signals coming from faraway galaxies. How can they be sure that a particular radio signal was deliberately sent by intelligent beings instead of coming from some natural phenomenon? What if that signal began with a series of beeps in a pattern comprised of the first 30 prime numbers (“beep-beep,” “beep-beep-beep,” and so on)?
TEACHING TIPAn alternative definitionis that a prime number is a positive integer withexactly two different factors. You may want to point out that 2 is theonly even prime number.
LessonNotes
1 Focus1 Focus
Chapter 9 Resource Masters• Study Guide and Intervention, pp. 523–524• Skills Practice, p. 525• Practice, p. 526• Reading to Learn Mathematics, p. 527• Enrichment, p. 528
Lesson 9-1 Factors and Greatest Common Factors 475
When a whole number is expressed as the product of factors that are all primenumbers, the expression is called the of the number. prime factorization
A negative integer is factored completely when it is expressed as the product of �1 and prime numbers.
www.algebra1.com/extra_examples
b. 23
The only whole numbers that can be multiplied together to get 23 are 1 and 23.Therefore, the factors of 23 are 1 and 23. Since the only factors of 23 are 1 anditself, 23 is a prime number.
Prime Factorization of a Positive IntegerFind the prime factorization of 90.
Method 1
90 � 2 � 45 The least prime factor of 90 is 2.
� 2 � 3 � 15 The least prime factor of 45 is 3.
� 2 � 3 � 3 � 5 The least prime factor of 15 is 3.
All of the factors in the last row are prime. Thus, the prime factorization of 90 is 2 � 3 � 3 � 5.
Method 2
Use a factor tree.
90
9 � 10 90 � 9 � 10
3 � 3 � 2 � 5 9 � 3 � 3 and 10 � 2 � 5
All of the factors in the last branch of the factor tree are prime. Thus, the primefactorization of 90 is 2 � 3 � 3 � 5 or 2 � 32 � 5.Usually the factors are ordered from the least prime factor to the greatest.
Example 2Example 2
Prime Factorization of a Negative IntegerFind the prime factorization of �140.
�140 � �1 � 140 Express �140 as �1 times 140.
� �1 � 2 � 70 140 � 2 � 70
� �1 � 2 � 7 � 10 70 � 7 � 10
� �1 � 2 � 7 � 2 � 5 10 � 2 � 5
Thus, the prime factorization of �140 is �1 � 2 � 2 � 5 � 7 or �1 � 22 � 5 � 7.
Example 3Example 3
A monomial is in when it is expressed as the product of primenumbers and variables and no variable has an exponent greater than 1.
factored form
TEACHING TIPPoint out that studentscould have started with 2 � 45, 3 � 30, or 5 � 18,and found the same primefactorization.
Study Tip
UniqueFactorizationTheoremThe prime factorization ofevery number is uniqueexcept for the order inwhich the factors arewritten.
Prime NumbersBefore deciding that anumber is prime, trydividing it by all of theprime numbers that areless than the square rootof that number.
Lesson 9-1 Factors and Greatest Common Factors 475
Verbal/Linguistic Factoring integers is a very visual skill, whether it isdone by writing out factors in a line, or by using a factor tree. Havestudents describe one of the methods of factoring in their own words,without actually writing out the factors as an example. Alternatively, havestudents write a description of how to factor, using an example as aguide.
Factor each number. Thenclassify each number as primeor composite.
a. 22 The factors are 1, 2, 11, and22, so the number is composite.
b. 31 The factors are 1 and 31, sothe number is prime.
Teaching Tip Explain that theprime factorization for eachnumber is unique, so as long asyou divide by prime numbers,you will get the same primefactorization. For example:90 � 3 � 30
� 3 � 3 � 10� 3 � 3 � 2 � 5,
which is the same as 2 � 3 � 3 � 5that was found in Example 2.
Find the prime factorizationof 84. 2 � 2 � 3 � 7 or 22 � 3 � 7
Find the prime factorizationof �132. �1 � 22 � 3 � 11
Concept CheckPrime Factorization José andLatecia both found the primefactorization of 60. José got 22 � 3 � 5, and Latecia got 3 � 22 � 5.Explain which is correct. Both arecorrect. Each number has a uniqueprime factorization.
2 � 32 � x � x � x � y � y � yb. �26rst2 �1 � 2 � 13 � r � s � t � t
Teaching Tip Remind studentsthat prime factorization of a con-stant can have exponents greaterthan 1, but prime factorizationof variable values cannot.
GREATEST COMMONFACTOR
Find the GCF of each set ofmonomials.
a. 12 and 18 The GCF is 6
b. 27a2b and 15ab2c The GCF is 3ab
CRAFTS Rene has crocheted32 squares for an afghan. Eachsquare is 1 foot square. She isnot sure how she will arrangethe squares but does know itwill be rectangular and havea ribbon trim. What is themaximum amount of ribbonshe might need to finish theafghan? 66 ft
• The GCF of two or more integers is the product of the prime factors common tothe integers.
• The GCF of two or more monomials is the product of their common factors wheneach monomial is in factored form.
• If two or more integers or monomials have a GCF of 1, then the integers ormonomials are said to be relatively prime.
Greatest Common Factor (GCF)
GREATEST COMMON FACTOR Two or more numbers may have somecommon prime factors. Consider the prime factorization of 48 and 60.
48 � 2 � 2 � 2 � 2 � 3 Factor each number.
60 � 2 � 2 � 3 � 5 Circle the common prime factors.
The integers 48 and 60 have two 2s and one 3 as common prime factors. The productof these common prime factors, 2 � 2 � 3 or 12, is called the
of 48 and 60. The GCF is the greatest number that is a factor of both originalnumbers.(GCF)
greatest common factor
476 Chapter 9 Factoring
Prime Factorization of a MonomialFactor each monomial completely.
a. 12a2b3
12a2b3 � 2 � 6 � a � a � b � b � b 12 � 2 � 6, a2 � a � a, and b3 � b � b � b
� 2 � 2 � 3 � a � a � b � b � b 6 � 2 � 3
Thus, 12a2b3 in factored form is 2 � 2 � 3 � a � a � b � b � b.
b. �66pq2
�66pq2 � �1 � 66 � p � q � q Express �66 as �1 times 66.
� �1 � 2 � 33 � p � q � q 66 � 2 � 33
� �1 � 2 � 3 � 11 � p � q � q 33 � 3 � 11
Thus, �66pq2 in factored form is �1 � 2 � 3 � 11 � p � q � q.
Example 4Example 4
GCF of a Set of MonomialsFind the GCF of each set of monomials.
a. 15 and 16
15 � 3 � 5 Factor each number.
16 � 2 � 2 � 2 � 2 Circle the common prime factors, if any.
There are no common prime factors, so the GCF of 15 and 16 is 1. This means that 15 and 16 are relatively prime.
b. 36x2y and 54xy2z
36x2y � 2 � 2 � 3 � 3 � x � x � y Factor each number.
54xy2z � 2 � 3 � 3 � 3 � x � y � y � z Circle the common prime factors.
The GCF of 36x2y and 54xy2z is 2 � 3 � 3 � x � y or 18xy.
Example 5Example 5
AlternativeMethodYou can also find thegreatest common factorby listing the factors ofeach number and findingwhich of the commonfactors is the greatest.Consider Example 5a.
15: 1, 3, 5, 1516: 1, 2, 4, 8, 16
The only common factor,and therefore, the greatestcommon factor, is 1.
Study Tip
476 Chapter 9 Factoring
PowerPoint®
PowerPoint®
InteractiveChalkboard
PowerPoint®
Presentations
This CD-ROM is a customizableMicrosoft® PowerPoint®presentation that includes:• Step-by-step, dynamic solutions of
each In-Class Example from theTeacher Wraparound Edition
• Additional, Your Turn exercises foreach example
• The 5-Minute Check Transparencies• Hot links to Glencoe Online
Study Tools
Lisa Cook Kaysville Jr. H.S., Kaysville, UT
“To help in identifying prime factors, I like to have my students explore the Sieveof Eratosthenes. Use a 10-by-10 grid with the numbers 1-100 on it. Cross out 1since prime numbers are greater than 1. Circle 2 and cross out all multiples of 2.Circle 3 and cross out multiples of 3. Continue with the next odd number untilall multiples have been eliminated. The circled numbers are the prime numbersless than 100."
Teacher to TeacherTeacher to Teacher
Practice and ApplyPractice and Apply
indicates increased difficulty�
1. Determine whether the following statement is true or false. If false, provide acounterexample. All prime numbers are odd. false; 2
2. Explain what it means for two numbers to be relatively prime. Their GCF is 1.3. OPEN ENDED Name two monomials whose GCF is 5x2.
Sample answer: 5x2 and 10x3
Find the factors of each number. Then classify each number as prime or composite.
19. GARDENING Ashley is planting 120 tomato plants in her garden. In what wayscan she arrange them so that she has the same number of plants in each row, atleast 5 rows of plants, and at least 5 plants in each row? 5 rows of 24 plants,6 rows of 20 plants, 8 rows of 15 plants, or 10 rows of 12 plants
3 � 13 � b � b � b � c � c
1, 2, 4, 7, 8, 14,16, 28, 56, 112;composite
Concept Check
Guided Practice
12. �1 � 2 � 2 � 5 � 5 �x � x � x � y � z � z
Application
Lesson 9-1 Factors and Greatest Common Factors 477www.algebra1.com/self_check_quiz
Use FactorsGEOMETRY The area of a rectangle is 28 square inches. If the length and widthare both whole numbers, what is the maximum perimeter of the rectangle?
Find the factors of 28, and draw rectangles with each length and width. Then findeach perimeter.
The factors of 28 are 1, 2, 4, 7, 14, and 28.
The greatest perimeter is 58 inches. The rectangle with this perimeter has a lengthof 28 inches and a width of 1 inch.
Example 6Example 6
P � 1 � 28 � 1 � 28 or 58
1
28
14
2
P � 2 � 14 � 2 � 14 or 32
7
4
P � 4 � 7 � 4 � 7 or 22
GUIDED PRACTICE KEYExercises Examples
4–6 17–9 2, 3
10–12 413–18 5
19 6
Find the factors of each number. Then classify each number as prime or composite.
20. 19 21. 25 22. 80 23. 61
24. 91 25. 119 26. 126 27. 304
20–27. See margin.
Lesson 9-1 Factors and Greatest Common Factors 477
3 Practice/Apply3 Practice/Apply
Study NotebookStudy NotebookHave students—• add the definitions/examples of
the vocabulary terms to theirVocabulary Builder worksheets forChapter 9.
• include explanations on how tofactor numbers, find primefactorizations, and find thegreatest common factor.
• include any other item(s) that theyfind helpful in mastering the skillsin this lesson.
About the Exercises…Organization by Objective• Prime Factorization: 20–27,
32–39• Greatest Common Factor:
48–61
Odd/Even AssignmentsExercises 20–29 and 32–61 arestructured so that studentspractice the same conceptswhether they are assignedodd or even problems.
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
Less
on
9-1
Prime Factorization When two or more numbers are multiplied, each number is calleda factor of the product.
Definition Example
Prime NumberA prime number is a whole number, greater than 1, whose only factors are
51 and itself.
Composite NumberA composite number is a whole number, greater than 1, that has more than
10two factors.
Prime FactorizationPrime factorization occurs when a whole number is expressed as a product
45 � 32 � 5of factors that are all prime numbers.
Factor each number.Then classify each number as prime orcomposite.
a. 28To find the factors of 28, list all pairs ofwhole numbers whose product is 28.1 � 28 2 � 14 4 � 7Therefore, the factors of 28 are 1, 2, 4, 7,14, and 28. Since 28 has more than 2factors, it is a composite number.
b. 31To find the factors of 31, list all pairs ofwhole numbers whose product is 31.1 � 31Therefore, the factors of 31 are 1 and 31.Since the only factors of 31 are itself and1, it is a prime number.
Find the primefactorization of 200.Method 1200 � 2 � 100
� 2 � 2 � 50� 2 � 2 � 2 � 25� 2 � 2 � 2 � 5 � 5
All the factors in the last row are prime, sothe prime factorization of 200 is 23 � 52.
Method 2Use a factor tree.
All of the factors in each last branch of thefactor tree are prime, so the primefactorization of 200 is 23 � 52.
2 � 2 � 5 � 2 � 5
2 � 10 � 10
2 � 100
200
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the factors of each number. Then classify the number as prime or composite.
GEOMETRY For Exercises 31 and 32, use the following information.The area of a rectangle is 84 square inches. Its length and width are both whole numbers.
31. What is the minimum perimeter of the rectangle? 38 in.
32. What is the maximum perimeter of the rectangle? 170 in.
RENOVATION For Exercises 33 and 34, use the following information.Ms. Baxter wants to tile a wall to serve as a splashguard above a basin in the basement.She plans to use equal-sized tiles to cover an area that measures 48 inches by 36 inches.
33. What is the maximum-size square tile Ms. Baxter can use and not have to cut any of the tiles? 12-in. square
34. How many tiles of this size will she need? 12
Practice (Average)
Factors and Greatest Common Factors
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1Skills Practice, p. 525 and Practice, p. 526 (shown)
Reading to Learn Mathematics
Factors and Greatest Common Factors
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
Pre-Activity How are prime numbers related to the search for extraterrestrial life?
Read the introduction to Lesson 9-1 at the top of page 474 in your textbook.
If each “beep” counts as one, what are the first two prime numbers?
2 and 3
Reading the Lesson
1. Every whole number greater than 1 is either composite or .
2. Complete each statement.
a. In the prime factorization of a whole number, each factor is a number.
b. In the prime factorization of a negative integer, all the factors are prime except the
factor .
3. Explain why the monomial 5x2y is not in factored form.
The variable x has an exponent that is greater than 1.
4. Explain the steps used below to find the greatest common factor (GCF) of 84 and 120.
84 � 2 � 2 � 3 � 7 Write the prime factorization of 84.
120 � 2 � 2 � 2 � 3 � 5 Write the prime factorization of 120.
Common prime factors: 2, 2, 3 Identify common prime factors of 84 and 120.
2 � 2 � 3 � 12 Multiply the common factors to find the GCF of 84 and 120.
Helping You Remember
5. How can the two words that make up the term prime factorization help you rememberwhat the term means?
Sample answer: The word factorization reminds you that the numbermust be expressed as a product of factors. The word prime reminds youthat with the possible exception of �1, all of the factors used must beprime numbers.
�1
prime
prime
Reading to Learn Mathematics, p. 527
Finding the GCF by Euclid’s AlgorithmFinding the greatest common factor of two large numbers can take a long time using prime factorizations. This method can be avoided by using Euclid’s Algorithm as shown in the following example.
Find the GCF of 209 and 532.
Divide the greater number, 532, by the lesser, 209.
2209�5�3�2�
418 1114 �2�0�9�
114 195 �1�1�4�
95 519 �9�5�
950
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-19-1
ExampleExample
Divide the remainderinto the divisor above.Repeat this processuntil the remainderis zero. The lastnonzero remainder is the GCF.
Enrichment, p. 528
478 Chapter 9 Factoring
Homework HelpFor See
Exercises Examples20–27, 62, 1
65, 6632–39 2, 340–47 448–61, 563, 64
28–31, 67 6
Extra PracticeSee page 839.
Marching BandsDrum Corps International(DCI) is a nonprofit youthorganization serving juniordrum and bugle corpsaround the world.Members of thesemarching bands rangefrom 14 to 21 years of age.Source: www.dci.org
�
GEOMETRY For Exercises 28 and 29, consider a rectangle whose area is 96 square millimeters and whose length and width are both whole numbers.
28. What is the minimum perimeter of the rectangle? Explain your reasoning. 40 mm29. What is the maximum perimeter of the rectangle? Explain your reasoning.
194 mm 28–29. See margin for explanations.
COOKIES For Exercises 30 and 31, use the following information.A bakery packages cookies in two sizes of boxes, one with 18 cookies and the otherwith 24 cookies. A small number of cookies are to be wrapped in cellophane beforethey are placed in a box. To save money, the bakery will use the same sizecellophane packages for each box.
30. How many cookies should the bakery place in each cellophane package tomaximize the number of cookies in each package? 6 cookies
31. How many cellophane packages will go in each size box?3 packages in the box of 18 cookies and 4 packages in the box of 24 cookies.
62. NUMBER THEORY Twin primes are two consecutive odd numbers that areprime. The first pair of twin primes is 3 and 5. List the next five pairs of twinprimes. 5, 7; 11, 13; 17, 19; 29, 31; 41, 43
MARCHING BANDS For Exercises 63 and 64, use the following information.Central High’s marching band has 75 members, and the band from Northeast Highhas 90 members. During the halftime show, the bands plan to march into thestadium from opposite ends using formations with the same number of rows.
63. If the bands want to match up in the center of the field, what is the maximumnumber of rows? 15
64. How many band members will be in each row after the bands are combined? 11
NUMBER THEORY For Exercises 65 and 66, use the following information.One way of generating prime numbers is to use the formula 2p � 1, where p is aprime number. Primes found using this method are called Mersenne primes. Forexample, when p � 2, 22 � 1 � 3. The first Mersenne prime is 3.
65. Find the next two Mersenne primes. 7, 3166. Will this formula generate all possible prime numbers? Explain your reasoning.
No, it does not generate the first prime number, 2.Online Research Data Update What is the greatest known primenumber? Visit www.algebra1.com/data_update to learn more.
478 Chapter 9 Factoring
ELL
Answers
28. The factors of 96 whose sum whendoubled is the least are 12 and 18.
29. The factors of 96 whose sum whendoubled is the greatest are 1 and 96.
Modeling Write a number andeach step of its prime factorization(using the factor tree method) onnote cards. Each factor, includingthe intermediate factors, shouldbe on separate note cards. Thendraw some arrows on other notecards. Tape the cards at randomon the chalkboard. Have studentvolunteers come up and arrangethe factors to find the primefactorization of the number.
Getting Ready for Lesson 9-2PREREQUISITE SKILL Studentswill learn to factor polynomialsusing the Distributive Propertyin Lesson 9-2. Students shouldknow how to rewrite expressionsusing the Distributive Propertyso they can factor polynomials.Use Exercises 81–86 to determineyour students’ familiarity withthe Distributive Property.
Answers
40. 2 � 3 � 11 � d � d � d � d
41. 5 � 17 � x � x � y � y
42. 7 � 7 � a � a � a � b � b
43. 2 � 5 � 5 � g � h
44. 2 � 2 � 2 � 2 � 2 � 2 � 2 � p � q � q
45. 3 � 3 � 3 � 3 � 3 � n � n � n � m
46. �1 � 3 � 61 � x � y � z � z � z
47. �1 � 13 � 13 � a � a � b � c � c
67. base 1 cm, height 40 cm; base 2 cm, height 20 cm; base 4 cm,height 10 cm; base 5 cm, height 8 cm; base 8 cm, height 5 cm;base 10 cm, height 4 cm; base 20 cm, height 2 cm; base 40 cm,height 1 cm
68b.If 6 is a factor of ab, then theprime factorization of ab mustcontain 2 � 3 So, 3 must be afactor of either a or b.
Finding the GCF of
distances will help you
make a scale model of
the solar system. Visit
www.algebra1.com/webquest to continue
work on your WebQuest
project.
Lesson 9-1 Factors and Greatest Common Factors 479
Getting Ready forthe Next Lesson
Maintain Your SkillsMaintain Your Skills
67. GEOMETRY The area of a triangle is 20 square centimeters. What are possiblewhole-number dimensions for the base and height of the triangle? See margin.
68. CRITICAL THINKING Suppose 6 is a factor of ab, where a and b are naturalnumbers. Make a valid argument to explain why each assertion is true orprovide a counterexample to show that an assertion is false.
a. 6 must be a factor of a or of b.
b. 3 must be a factor of a or of b. True; see margin for explanation.c. 3 must be a factor of a and of b. false; counterexample: a � 3, b � 1082
69. Answer the question that was posed at the beginning ofthe lesson. See p. 521A.
How are prime numbers related to the search for extraterrestrial life?
Include the following in your answer:
• a list of the first 30 prime numbers and an explanation of how you foundthem, and
• an explanation of why a signal of this kind might indicate that anextraterrestrial message is to follow.
70. Miko claims that there are at least four ways to design a 120-square-footrectangular space that can be tiled with 1-foot by 1-foot tiles. Which statementbest describes this claim? D
Her claim is false because 120 is a prime number.
Her claim is false because 120 is not a perfect square.
Her claim is true because 240 is a multiple of 120.
Her claim is true because 120 has at least eight factors.
71. Suppose Ψx is defined as the largest prime factor of x. For which of thefollowing values of x would Ψx have the greatest value? A
Find the value of r so that the line that passes through the given points has thegiven slope. (Lesson 5-1)
78. (1, 2), (�2, r), m � 3 �7 79. (�5, 9), (r, 6), m � ��35
� 0
80. RETAIL SALES A department store buys clothing at wholesale prices and thenmarks the clothing up 25% to sell at retail price to customers. If the retail price ofa jacket is $79, what was the wholesale price? (Lesson 3-7) $63.20
PREREQUISITE SKILL Use the Distributive Property to rewrite each expression.(To review the Distributive Property, see Lesson 1-5.)
Sometimes you know the product of binomials and are asked to find the factors. This is called factoring. You can use algebra tiles and a product mat to factor binomials.
Factoring Using the Distributive Property
Model and AnalyzeUse algebra tiles to factor each binomial.1. 2x � 10 2(x � 5) 2. 6x � 8 2(3x � 4) 3. 5x2 � 2x x(5x � 2) 4. 9 � 3x 3(3 � x)Tell whether each binomial can be factored. Justify your answer with a drawing.5. 4x � 10 yes 6. 3x � 7 no 7. x2 � 2x yes 8. 2x2 � 3 no5–8. See pp. 521A–521B for drawings.9. MAKE A CONJECTURE Write a paragraph that explains how you can use algebra tiles to
determine whether a binomial can be factored. Include an example of one binomial that can be factored and one that cannot.
Activity 1 Use algebra tiles to factor 3x � 6.
Model the polynomial 3x � 6. Arrange the tiles into a rectangle.
The total area of the rectangle
represents the product, and its length
and width represent the factors.
The rectangle has a width of 3 and a
length of x � 2. So, 3x � 6 � 3(x � 2).
3
x � 2
x 1 1x 1 1x 1 1
1
1
1
1
1
1
x x x
Model the polynomial x2 � 4x. Arrange the tiles into a rectangle.
The rectangle has a width of x and a
length of x � 4. So, x2 � 4x � x(x � 4).
x �x �x �x �xx 2
x � 4
�x �x �x �xx 2
Activity 2 Use algebra tiles to factor x2 � 4x.
9. Binomials can be factored if they can be represented by a rectangle.Examples: 2x � 2 can be factored and 2x � 1 cannot be factored.
480 Chapter 9 Factoring
Teaching Algebra withManipulatives• pp. 10–11 (master for algebra tiles)• p. 17 (master for product mat)• p. 156 (student recording sheet)
Glencoe Mathematics Classroom Manipulative Kit• algebra tiles• product mat
Resource ManagerResource Manager
Objective Factor polynomialswith algebra tiles.
Materialsalgebra tilesproduct mat
• Remind students that they usedrectangles at the beginning ofLesson 9-1 to find factors ofwhole numbers. The procedurefor finding the factors ofpolynomials with algebra tilesis very similar. The length andwidth of a modeled polynomialrepresent the factors of thepolynomial.
• In Exercises 1–4, students needto recognize that they mustarrange the tiles into a rectanglewith a width greater than onein order to find the factors.
• In Exercises 5–9, studentsshould recognize that thebinomials that cannot be fac-tored can only be modeled in arectangle with a width of one.
You may wish to have studentssummarize this activity and whatthey learned from it.
5-Minute CheckTransparency 9-2 Use as a
quiz or review of Lesson 9-1.
Mathematical Background notesare available for this lesson on p. 472C.
Building on PriorKnowledge
In Chapter 1, students wereintroduced to the DistributiveProperty and learned how to useit to simplify expressions. InChapter 8, students learned tomultiply a polynomial by amonomial using the DistributiveProperty. In this lesson, studentswill reverse that process to factorpolynomials.
can you determine howlong a baseball will
remain in the air?Ask students:• What is the greatest common
factor (GCF) of two numbers?The GCF is the greatest numberthat is a factor of both originalnumbers.
• What is the GCF of 151t and16t2? t
• What is the height of the ballwhen t � 0? 0 ft
• What is the height of the ballwhen t � 1? 135 ft
FACTOR BY USING THE DISTRIBUTIVE PROPERTY In Chapter 8, youused the Distributive Property to multiply a polynomial by a monomial.
2a(6a � 8) � 2a(6a) � 2a(8)� 12a2 � 16a
You can reverse this process to express a polynomial as the product of a monomialfactor and a polynomial factor.
12a2 � 16a � 2a(6a) � 2a(8)� 2a(6a � 8)
Thus, a factored form of 12a2 � 16a is 2a(6a � 8).
a polynomial means to find its completely factored form. The expression2a(6a � 8) is not completely factored since 6a � 8 can be factored as 2(3a � 4).
Factoring
Factoring Using theDistributive Property
Lesson 9-2 Factoring Using the Distributive Property 481
can you determine how long a baseball will remain in the air?can you determine how long a baseball will remain in the air?
Use the Distributive PropertyUse the Distributive Property to factor each polynomial.
a. 12a2 � 16a
First, find the GCF of 12a2 and 16a.
12a2 � 2 � 2 � 3 � a � a Factor each number.
16a � 2 � 2 � 2 � 2 � a Circle the common prime factors.
GCF: 2 • 2 • a or 4a
Write each term as the product of the GCF and its remaining factors. Then usethe Distributive Property to factor out the GCF.
12a2 � 16a � 4a(3 � a) � 4a(2 � 2) Rewrite each term using the GCF.
� 4a(3a) � 4a(4) Simplify remaining factors.
� 4a(3a � 4) Distributive Property
Thus, the completely factored form of 12a2 � 16a is 4a(3a � 4).
Example 1Example 1
Nolan Ryan, the greatest strike-out pitcher in the history of baseball, had a fastball clocked at 98 miles per hour or about 151 feet per second. If he threw a ball directly upward with the same velocity, the height h of the ball in feet above the point at which he released it could be modeled by the formula h � 151t � 16t2, where t is the time in seconds. You can use factoring and the Zero Product Property to determine how long the ball would remain in the air before returning to his glove.
• Factor polynomials by using the Distributive Property.
• Solve quadratic equations of the form ax2 � bx � 0.
Vocabulary• factoring• factoring by grouping
Look BackTo review the DistributiveProperty, see Lesson 1-5.
Study Tip
Lesson x-x Lesson Title 481
Chapter 9 Resource Masters• Study Guide and Intervention, pp. 529–530• Skills Practice, p. 531• Practice, p. 532• Reading to Learn Mathematics, p. 533• Enrichment, p. 534• Assessment, p. 573
Parent and Student Study GuideWorkbook, p. 69
Prerequisite Skills Workbook, pp. 13–14School-to-Career Masters, p. 17
Teaching Tip Tell students thatone way to find the remainingfactors is to divide each term bythe GCF.
Use the Distributive Propertyto factor each polynomial.
a. 15x � 25x2 5x(3 � 5x)
b. 12xy � 24xy2 � 30x2y4
6xy(2 � 4y � 5xy3)
Teaching Tip Remind studentsthat when using the FOILmethod, they multiply the Firstterms, Outer terms, Inner terms,and Last terms.
Factor 2xy � 7x � 2y � 7.(x � 1)(2y � 7)
Factor 15a � 3ab � 4b � 20.(�3a � 4)(b � 5)
Concept CheckFactoring Using theDistributive Property Malcolmand Fatima each factored thepolynomial 2ax � 6cx � ab � 3bc.Malcolm’s answer was (2x � b)(a � 3c) and Fatima’swas (a � 3c)(2x � b). Which iscorrect? Explain your answer.Both are correct. The order in whichfactors are multiplied does not affectthe product.
The Distributive Property can also be used to factor some polynomials havingfour or more terms. This method is called because pairs ofterms are grouped together and factored. The Distributive Property is then applied a second time to factor a common binomial factor.
factoring by grouping
482 Chapter 9 Factoring
Use GroupingFactor 4ab � 8b � 3a � 6.
4ab � 8b � 3a � 6
� (4ab � 8b) � (3a � 6) Group terms with common factors.
� 4b(a � 2) � 3(a � 2) Factor the GCF from each grouping.
Factoring byGroupingSometimes you can groupterms in more than one waywhen factoring a polynomial.For example, the polynomialin Example 2 could havebeen factored in thefollowing way.4ab � 8b � 3a � 6� (4ab � 3a) � (8b � 6)� a(4b � 3) � 2(4b � 3)� (4b � 3)(a � 2)
Notice that this result is thesame as in Example 2.
Study Tip
FactoringTrinomialsSince the order in whichfactors are multiplied doesnot affect the product, (�5x � 3)(y � 7) is alsoa correct factoring of 35x � 5xy � 3y � 21.
Study Tip
Factoring by Grouping • Words A polynomial can be factored by grouping if all of the following
situations exist.• There are four or more terms.• Terms with common factors can be grouped together.• The two common factors are identical or are additive inverses of
each other.
• Symbols ax � bx � ay � by � x(a � b) � y(a � b)� (a � b)(x � y)
b. 18cd2 � 12c2d � 9cd
18cd2 � 2 � 3 � 3 � c � d � d Factor each number.
12c2d � 2 � 2 � 3 � c � c � d Circle the common prime factors.
9cd � 3 � 3 � c � dGCF: 3 � c � d or 3cd
18cd2 � 12c2d � 9cd � 3cd(6d) � 3cd(4c) � 3cd(3) Rewrite each term using the GCF.
� 3cd(6d � 4c � 3) Distributive Property
Recognizing binomials that are additive inverses is often helpful when factoring by grouping. For example, 7 � y and y � 7 are additive inverses because their sum is 0. By rewriting 7 � y in the factored form �1(y � 7), factoring by grouping ismade possible in the following example.
Use the Additive Inverse PropertyFactor 35x � 5xy � 3y � 21.
35x � 5xy � 3y � 21 � (35x � 5xy) � (3y � 21) Group terms with common factors.
� 5x(7 � y) � 3(y � 7) Factor the GCF from each grouping.
� 5x(�1)(y � 7) � 3(y � 7) 7 � y � �1(y � 7)
� �5x(y � 7) � 3(y � 7) 5x(�1) � �5x
� (y � 7)(�5x � 3) Distributive Property
Example 3Example 3
482 Chapter 9 Factoring
2 Teach2 Teach
PowerPoint®
Zero Product Property • Words If the product of two factors is 0, then at least one of the factors
must be 0.
• Symbols For any real numbers a and b, if ab � 0, then either a � 0, b � 0, or both a and b equal zero.
Lesson 9-2 Factoring Using the Distributive Property 483
SOLVE EQUATIONS BY FACTORING Some equations can be solved byfactoring. Consider the following products.
6(0) � 0 0(�3) � 0 (5 � 5)(0) � 0 �2(�3 � 3) � 0
Notice that in each case, at least one of the factors is zero. These examples illustratethe .Zero Product Property
Solve an Equation in Factored FormSolve (d � 5)(3d � 4) � 0. Then check the solutions.
If (d � 5)(3d � 4) � 0, then according to the Zero Product Property either d � 5 � 0 or 3d � 4 � 0.
(d � 5)(3d � 4) � 0 Original equation
d � 5 � 0 or 3d � 4 � 0 Set each factor equal to zero.
d � 5 3d � �4 Solve each equation.
d � ��43
�
The solution set is �5, ��43
��.
CHECK Substitute 5 and ��43
� for d in the original equation.
(d � 5)(3d � 4) � 0 (d � 5)(3d � 4) � 0
(5 � 5)[3(5) � 4] � 0 ��43
� � 5�3��43
� � 4� � 0
(0)(19) � 0 ��139�(0) � 0
0 � 0 � 0 � 0 �
Example 4Example 4
CommonMisconception You may be tempted totry to solve the equationin Example 5 by dividingeach side of the equationby x. Remember, however,that x is an unknownquantity. If you divide byx, you may actually bedividing by zero, which isundefined.
Study Tip Solve an Equation by FactoringSolve x2 � 7x. Then check the solutions.
Write the equation so that it is of the form ab � 0.
x2 � 7x Original equation
x2 � 7x � 0 Subtract 7x from each side.
x(x � 7) � 0 Factor the GCF of x2 and �7x, which is x.
x � 0 or x � 7 � 0 Zero Product Property
x � 7 Solve each equation.
The solution set is {0, 7}. Check by substituting 0 and 7 for x in the original equation.
Example 5Example 5
If an equation can be written in the form ab � 0, then the Zero Product Propertycan be applied to solve that equation.
www.algebra1.com/extra_examples
Lesson 9-2 Factoring Using the Distributive Property 483
Visual/Spatial When students solve factored equations, have themwrite each factor on a separate sheet of scrap paper, followed by “� 0”on a third sheet of scrap paper. Then, have students remove one of thefactors and solve the remaining equation. Once the first solution isfound, have students place the other factor equal to zero and solve forthe second solution.
Differentiated Instruction
44
55
In-Class ExamplesIn-Class Examples
SOLVE EQUATIONS BY FACTORING
Teaching Tip If students thinkusing the Zero Product Propertyto set factors equal to zero issomewhat arbitrary, have themmultiply the two factors toobtain a polynomial, and set thepolynomial equal to zero. Thenhave students substitute the twosolutions into the polynomial tosee that they produce a truesentence.
�2, �Teaching Tip Remind studentsthat for the Zero ProductProperty to work, one of twofactors is equal to zero.Therefore, students must factorx2 � 7x before they can assumethat one term is equal to zero.
Study NotebookStudy NotebookHave students—• add the definitions/examples of
the vocabulary terms to theirVocabulary Builder worksheets forChapter 9.
• include explanations on how tofactor polynomials using thedistributive property, and how tosolve equations using factoring.
• include any other item(s) that theyfind helpful in mastering the skillsin this lesson.
484 Chapter 9 Factoring
Practice and ApplyPractice and Apply
indicates increased difficulty�
1. Write 4x2 � 12x as a product of factors in three different ways. Then decide whichof the three is the completely factored form. Explain your reasoning. See margin.
2. OPEN ENDED Give an example of the type of equation that can be solved byusing the Zero Product Property.
3. Explain why (x � 2)(x � 4) � 0 cannot be solved by dividing each side by x � 2.The division would eliminate 2 as a solution.
PHYSICAL SCIENCE For Exercises 13–15, use the information below and in the graphic.
A flare is launched from a life raft. The height h ofthe flare in feet above the sea is modeled by theformula h � 100t � 16t2, where t is the time inseconds after the flare is launched.
13. At what height is the flare when it returns tothe sea? 0 ft
14. Let h � 0 in the equation h � 100t � 16t2 andsolve for t. 0, 6.25
15. How many seconds will it take for the flare to return to the sea? Explain your reasoning.6.25 s; The answer 0 is not reasonable sinceit represents the time at which the flare islaunched.
h � 100t � 16t2
100 ft /s
h � 0
Factor each polynomial. 16–39. See p. 521A.16. 5x � 30y 17. 16a � 4b 18. a5b � a
1. Sample answers: 4(x2 � 3x), x(4x � 12), or 4x(x � 3); 4x(x � 3); 4x is the GCF of 4x2
and 12x.
63. Answers should include the following.
• Let h � 0 in the equation h � 151t � 16t2. To solve 0 � 151t � 16t2, factor the right-hand side as t(151 � 16t). Then, since t(151 � 16t) � 0, either t � 0 or 151 � 16t � 0. Solving each equation for t, we find that t � 0 or t � 9.44.
• The solution t � 0 represents the point at which the ball was initially thrown into the air.The solution t � 9.44 represents how long it took after the ball was thrown for it to returnto the same height at which it was thrown.
Study Guide and Intervention
Factoring Using the Distributive Property
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
Less
on
9-2
Factor by Using the Distributive Property The Distributive Property has beenused to multiply a polynomial by a monomial. It can also be used to express a polynomial infactored form. Compare the two columns in the table below.
�0, � �� , 0� �� , 0�LANDSCAPING For Exercises 31 and 32, use the following information.A landscaping company has been commissioned to design a triangular flower bed for a mallentrance. The final dimensions of the flower bed have not been determined, but the companyknows that the height will be two feet less than the base. The area of the flower bed can be represented by the equation A � b2 � b.
31. Write this equation in factored form. A � b� b � 1�32. Suppose the base of the flower bed is 16 feet. What will be its area? 112 ft2
33. PHYSICAL SCIENCE Mr. Alim’s science class launched a toy rocket from ground levelwith an initial upward velocity of 60 feet per second. The height h of the rocket in feetabove the ground after t seconds is modeled by the equation h � 60t � 16t2. How longwas the rocket in the air before it returned to the ground? 3.75 s
1�2
1�2
13�4
3�2
5�6
1�2
4�3
5�2
7�3
Practice (Average)
Factoring Using the Distributive Property
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2Skills Practice, p. 531 and Practice, p. 532 (shown)
Reading to Learn Mathematics
Factoring Using the Distributive Property
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2
Less
on
9-2
Pre-Activity How can you determine how long a baseball will remain in the air?
Read the introduction to Lesson 9-2 at the top of page 481 in your textbook.
In the formula h � 151t � 16t2, what does the number 151 represent?
the velocity of the baseball in feet per second at the instant it is thrown
Reading the Lesson
1. Factoring a polynomial means to find its completely factored form.
a. The expression x(6x � 9) is a factored form of the polynomial 6x2 � 9x. Why is thisnot its completely factored form?
The numbers 6 and 9 have a common factor of 3, which needs to befactored out of the expression.
b. Provide an example of a completely factored polynomial.
Sample answer: 2b(5b � 4) is the completely factored form of 10b2 � 8b.
c. Provide an example of a polynomial that is not completely factored.
Sample answer: b(10b � 8) is a factored form of 10b2 � 8b, but is notthe completely factored form.
2. The polynomial 5ab � 5b2 � 3a � 6b can be rewritten as 5b(a � b) � 3(a � 2b). Doesthis indicate that the original polynomial can be factored by grouping? Explain.
No, because 5b(a � b) and 3(a � 2b) do not have a common factor otherthan 1.
3. The polynomial 3x2 � 3xy � 2x � 2y can be rewritten as 3x(x � y) � 2(x � y). Does thisindicate that the original polynomial can be factored by grouping? Explain.
Yes, because 3x(x � y) and 2(x � y) have the common factor (x � y).
Helping You Remember
4. How would you explain to a classmate when it is possible to use the Zero ProductProperty to solve an equation?
The equation must have 0 on one side, and the other side must be theproduct of two expressions.
Reading to Learn Mathematics, p. 533
Perfect, Excessive, Defective, and Amicable NumbersA perfect number is the sum of all of its factors except itself.Here is an example:
28 � 1 � 2 � 4 � 7 � 14
There are very few perfect numbers. Most numbers are eitherexcessive or defective.
An excessive number is greater than the sum of all of its factorsexcept itself.
A defective number is less than this sum.
Two numbers are amicable if the sum of the factors of the firstnumber, except for the number itself, equals the second number,and vice versa.
Solve each problem.
1. Write the perfect numbers between 0 and 31.
6, 28
2 W it th i b b t 0 d 31
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-29-2Enrichment, p. 534
Lesson 9-2 Factoring Using the Distributive Property 485
SOFTBALL For Exercises 42 and 43, use the following information. Albertina is scheduling the games for a softball league. To find the number of
games she needs to schedule, she uses the equation g � �12
�n2 � �12
�n, where g
represents the number of games needed for each team to play each other teamexactly once and n represents the number of teams.
42. Write this equation in factored form. g � �12
�n(n � 1)43. How many games are needed for 7 teams to play each other exactly 3 times?
63 games
GEOMETRY Write an expression in factored form for the area of each shadedregion.
44. 45.
4(a � b � 4)
2r 2(4 � �)
GEOMETRY Find an expression for the area of a square with the given perimeter.
46. P � 12x � 20y in. 47. P � 36a � 16b cm9x2 � 30xy � 25y2 in2 81a2 � 72ab � 16b2 cm2
60. MARINE BIOLOGY In a pool at a water park, a dolphin jumps out of the water traveling at 20 feet per second. Its height h, in feet, above the water after t seconds is given by the formula h � 20t � 16t2. How long is the dolphin in the air before returning to the water? 1.25 s
61. BASEBALL Malik popped a ball straight up with an initial upward velocity of45 feet per second. The height h, in feet, of the ball above the ground is modeledby the equation h � 2 � 48t � 16t2. How long was the ball in the air if thecatcher catches the ball when it is 2 feet above the ground? about 2.8 s
62. CRITICAL THINKING Factor ax � y � axby � aybx � bx � y. (ax � bx)(ay � by)
63. Answer the question that was posed at the beginning ofthe lesson. See margin.
How can you determine how long a baseball will remain in the air?
Include the following in your answer:
• an explanation of how to use factoring and the Zero Product Property to findhow long the ball would be in the air, and
• an interpretation of each solution in the context of the problem.
WRITING IN MATH
r r �2
2
2
2a
b
www.algebra1.com/self_check_quiz
Marine BiologistMarine biologists studyfactors that affectorganisms living in andnear the ocean.
Online ResearchFor information about a career as a marinebiologist, visit:www.algebra1.com/careersSource: National Sea Grant Library
Lesson 9-2 Factoring Using the Distributive Property 485
Speaking Ask a volunteer todescribe the similarities and dif-ferences between factoring usinggrouping, and factoring usingthe additive inverse property.Encourage other students to askquestions.
Getting Ready for Lesson 9-3PREREQUISITE SKILL Studentswill learn to factor trinomials inLesson 9-3. It is important thatstudents recall how to multiplypolynomials to check that theycorrectly factored trinomials. UseExercises 76–81 to determineyour students’ familiarity withmultiplying polynomials.
Assessment Options
Practice Quiz 1 The quiz pro-vides students with a brief reviewof the concepts and skills inLessons 9-1 and 9-2. Lessonnumbers are given to the right ofthe exercises or instruction linesso students can review conceptsnot yet mastered.
Quiz (Lessons 9-1 and 9-2) isavailable on p. 573 of the Chapter 9Resource Masters.
Answer
1. 1, 3, 5, 9, 15, 25, 45, 75, 225;composite
486 Chapter 9 Factoring
Practice Quiz 1Practice Quiz 1
1. Find the factors of 225. Then classify the number as prime or composite. (Lesson 9-1) See margin.
2. Find the prime factorization of �320. (Lesson 9-1) �1 � 26 � 5
3. Factor 78a2bc3 completely. (Lesson 9-1) 2 � 3 � 13 � a � a � b � c � c � c
4. Find the GCF of 54x3, 42x2y, and 30xy2. (Lesson 9-1) 6x
64. The total number of feet in x yards, y feet, and z inches is A3x � y � �
1z2�. 12(x � y � z).
x � 3y � 36z. �3x6� � �
1
y2� � z.
65. QUANTITATIVE COMPARISON Compare the quantity in Column A and thequantity in Column B. Then determine whether: A
the quantity in Column A is greater,the quantity in Column B is greater,the two quantities are equal, orthe relationship cannot be determined from the information given.D
C
B
A
DC
BA
StandardizedTest Practice
Maintain Your SkillsMaintain Your Skills
Factor each number. Then classify each number as prime or composite. (Lesson 9-1)
Simplify. Assume that no denominator is equal to zero. (Lesson 8-2)
72. �ss�
4
7� s11 73. �1
1
8
2
xx
3
2
yy
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4
1
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1
3
7
4
(
pp
7
3
qq
2
rr�
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1
5
)2� �2r 3p�
75. FINANCE Michael uses at most 60% of his annual FlynnCo stock dividend topurchase more shares of FlynnCo stock. If his dividend last year was $885 andFlynnCo stock is selling for $14 per share, what is the greatest number of sharesthat he can purchase? (Lesson 6-2) 37 shares
PREREQUISITE SKILL Find each product.(To review multiplying polynomials, see Lesson 8-7.)
the negative solution of the negative solution of(a � 2)(a � 5) � 0 (b � 6)(b � 1) � 0
Column A Column B
Getting Ready forthe Next Lesson
76. n2 � 11n � 2477. x2 � 9x � 2078. b2 � 3b � 70
Lessons 9-1 and 9-2
486 Chapter 9 Factoring
4 Assess4 Assess
AlgebraActivity
Getting StartedGetting Started
TeachTeach
A Preview of Lesson 9-3
Objective Factor trinomials withalgebra tiles.
Materialsalgebra tilesproduct mat
• Remind students that they usedalgebra tiles in an activity priorto Lesson 9-2 to factor polyno-mials. Ask students to tell youwhat they must be able to formwith the tiles in order to factora polynomial. a rectangle
• Activity 1 Remind students toread the width of the tiles alongthe edge of the rectangle. Thex2 tile has a width of x and thex tiles have a width of one.
• Activity 2 Encourage studentsto try several different arrange-ments until they can form arectangle. While the x2 tileshould be in the corner, there ismore than one correct way toarrange the tiles into a rectangle.
Algebra Activity Factoring Trinomials 487
A Preview of Lesson 9-3
Factoring TrinomialsYou can use algebra tiles to factor trinomials. If a polynomial represents the area of a rectangleformed by algebra tiles, then the rectangle’s length and width are factors of the area.
(continued on the next page)
Activity 1 Use algebra tiles to factor x2 � 6x � 5.
Model the polynomial x2 � 6x � 5.
Place the x2 tile at the corner of the product mat. Arrange the 1 tiles into arectangular array. Because 5 is prime, the5 tiles can be arranged in a rectangle in one way, a 1-by-5 rectangle.
Complete the rectangle with the x tiles.
The rectangle has a width of x � 1 and a length of x � 5. Therefore, x2 � 6x + 5 � (x � 1)(x � 5).
x � 1
x � 5
1 1 1 1 1
x 2 x x x x x
x
1 1 1 1 1
x 2
x 21
1
1
1
1x x x x x x
Activity 2 Use algebra tiles to factor x2 � 7x � 6.
Model the polynomial x2 � 7x � 6.
Place the x2 tile at the corner of the product mat. Arrange the 1 tiles into arectangular array. Since 6 � 2 � 3, try a 2-by-3 rectangle. Try to complete therectangle. Notice that there are two extra x tiles.
1 1 1
1 1 1
x 2
x 2 x x x x x1
1
1
1
1
1
x x
Algebra Activity Factoring Trinomials 487
Teaching Algebra withManipulatives• pp. 10–11 (master for algebra tiles)• p. 17 (master for product mat)• p. 159 (student recording sheet)
Glencoe Mathematics Classroom Manipulative Kit• algebra tiles• product mat
Resource ManagerResource Manager
Study NotebookStudy Notebook
AssessAssess
488 Investigating Slope-Intercept Form
Algebra ActivityAlgebra Activity
488 Chapter 9 Factoring
Arrange the 1 tiles into a 1-by-6 rectangular
array. This time you can complete the
rectangle with the x tiles.
The rectangle has a width of x � 1
and a length of x � 6. Therefore,
x2 � 7x � 6 � (x � 1)(x � 6).
x � 1
x � 6
1 1 1 1 1 1
x 2 x x x x x x
x
Activity 3 Use algebra tiles to factor x2 � 2x � 3.
Model the polynomial x2 � 2x � 3.
Place the x2 tile at the corner of the product mat. Arrange the 1 tiles into a 1-by-3 rectangular array as shown.
Place the x tile as shown. Recall that you can add zero-pairs without changing thevalue of the polynomial. In this case, add a zero pair of x tiles.
The rectangle has a width of x � 1 and a length of x � 3. Therefore, x2 � 2x � 3 � (x � 1)(x � 3).
• Activity 3 Remind students topay close attention to the signof the tiles. It would be veryeasy to mistake the factors as(x � 3)(x � 3) if they do notpay attention to the signs.
• Students may need to bereminded that adding a zeropair is similar to adding thesame number to both sides ofan equation. Be sure thatstudents are careful to add onex tile and one �x tile whenthey add a zero pair.
• After students completeExercises 1–8, ask them whetherthey notice a correlationbetween the need to use zeropairs to factor the trinomial,and the appearance of theresulting factors. Sample answer:When zero pairs are used, thesigns of the factors are opposite.When zero pairs are not used, thesigns of the factors are the same.
You may wish to have studentssummarize this activity and whatthey learned from it.
5-Minute CheckTransparency 9-3 Use as a
quiz or review of Lesson 9-2.
Mathematical Background notesare available for this lesson on p. 472D.
can factoring be used to find the dimensions
of a garden?Ask students:• Why do you need to find two
numbers whose product is 54to find the dimensions of thegarden? The garden is a rectanglewith an area of 54 ft2. The area of arectangle is equal to the length timesthe width. Since the area is 54 ft2,the length and the width must betwo numbers whose product is 54.
• What two integers have aproduct of 54? 1 and 54, 2 and27, 3 and 18, 6 and 9
• Which pair has a sum of 15?6 and 9
• What are the dimensions of thevegetable garden? The garden is6 ft by 9 ft.
Factoring Trinomials: x2 � bx � c
can factoring be used to find the dimensions of a garden?can factoring be used to find the dimensions of a garden?
• Factor trinomials of the form x2 � bx � c.
• Solve equations of the form x2 � bx � c � 0.
Tamika has enough bricks to make a 30-footborder around the rectangular vegetable gardenshe is planting. The booklet she got from thenursery says that the plants will need a space of 54 square feet to grow. What should thedimensions of her garden be? To solve thisproblem, you need to find two numbers whoseproduct is 54 and whose sum is 15, half theperimeter of the garden.
A � 54 ft2
P � 30 ft
Reading MathA quadratic trinomial is a trinomial of degree 2.This means that thegreatest exponent of the variable is 2.
Study Tip
Factoring x2 � bx � c • Words To factor quadratic trinomials of the form x2 � bx � c, find
two integers, m and n, whose sum is equal to b and whose product isequal to c. Then write x2 � bx � c using the pattern (x � m)(x � n).
• Symbols x2 � bx � c � (x � m)(x � n) when m � n � b and mn � c.
• Example x2 � 5x � 6 � (x � 2)(x � 3), since 2 � 3 � 5 and 2 � 3 � 6.
FACTOR x2 � bx � c In Lesson 9-1, you learned that when two numbers aremultiplied, each number is a factor of the product. Similarly, when two binomialsare multiplied, each binomial is a factor of the product.
To factor some trinomials, you will use the pattern for multiplying two binomials.Study the following example.
F O I L(x � 2)(x � 3) � (x � x) � (x � 3) � (x � 2) � (2 � 3) Use the FOIL method.
� x2 � 3x � 2x � 6 Simplify.
� x2 � (3 � 2)x � 6 Distributive Property
� x2 � 5x � 6 Simplify.
Observe the following pattern in this multiplication.
(x � 2)(x � 3) � x2 � (3 � 2)x � (2 � 3)
(x � m)(x � n) � x2 � (n � m)x � mn
� x2 � (m � n)x � mn
x2 � bx � c b � m � n and c � mn
Notice that the coefficient of the middle term is the sum of m and n and the last termis the product of m and n. This pattern can be used to factor quadratic trinomials ofthe form x2 � bx � c.
� �
Lesson 9-3 Factoring Trinomials: x2 � bx � c 489
Lesson x-x Lesson Title 489
Chapter 9 Resource Masters• Study Guide and Intervention, pp. 535–536• Skills Practice, p. 537• Practice, p. 538• Reading to Learn Mathematics, p. 539• Enrichment, p. 540• Assessment, pp. 573, 575
To determine m and n, find the factors of c and use a guess-and-check strategy tofind which pair of factors has a sum of b.
490 Chapter 9 Factoring
b and c Are PositiveFactor x2 � 6x � 8.
In this trinomial, b � 6 and c � 8. You need to find two numbers whose sum is6 and whose product is 8. Make an organized list of the factors of 8, and look forthe pair of factors whose sum is 6.
Factors of 8 Sum of Factors
1, 8 92, 4 6 The correct factors are 2 and 4.
x2 � 6x � 8 � (x � m)(x � n) Write the pattern.
� (x � 2)(x � 4) m � 2 and n � 4
CHECK You can check this result by multiplying the two factors.
F O I L
(x � 2)(x � 4) � x2 � 4x � 2x � 8 FOIL method
� x2 � 6x � 8 � Simplify.
Example 1Example 1
Testing Factors Once you find the correctfactors, there is no needto test any other factors.Therefore, it is notnecessary to test �4 and�4 in Example 2.
Study Tip
b Is Negative and c Is PositiveFactor x2 � 10x � 16.
In this trinomial, b � �10 and c � 16. This means that m � n is negative and mn ispositive. So m and n must both be negative. Therefore, make a list of the negativefactors of 16, and look for the pair of factors whose sum is �10.
Factors of 16 Sum of Factors
�1, �16 �17�2, �8 �10�4, �4 �8
The correct factors are �2 and �8.
x2 � 10x � 16 � (x � m)(x � n) Write the pattern.
� (x � 2)(x � 8) m � �2 and n � �8
CHECK You can check this result by using a graphing calculator. Graph y � x2 � 10x � 16and y � (x � 2)(x � 8) on the same screen. Since only one graph appears, the two graphs must coincide. Therefore, the trinomial has been factored correctly. �
[�10, 10] scl: 1 by [�10, 10] scl: 1
Example 2Example 2
When factoring a trinomial where b is negative and c is positive, you can use what you know about the product of binomials to help narrow the list of possiblefactors.
You will find that keeping an organized list of the factors you have tested isparticularly important when factoring a trinomial like x2 � x � 12, where the valueof c is negative.
TEACHING TIPCaution students that twographs may appear tocoincide in the standardviewing window, but theydo not. Have them usethe TABLE feature to verifythe identical y values.
490 Chapter 9 Factoring
Kinesthetic As students are learning the rules for factoring trinomials,encourage them to use algebra tiles to confirm their results. Studentsshould soon realize that the greater the values of b and c in thetrinomials, the more cumbersome algebra tiles become, which shouldreinforce the importance of learning to factor using the method in the text.
Differentiated Instruction
2 Teach2 Teach
The concept offactoring trino-mials as intro-duced in thislesson may
seem somewhat abstract tosome students. Whenever youintroduce abstract concepts, itis good to reinforce them witha concrete example. After intro-ducing factoring trinomials, re-fer students back to the lessonopener problem. Ask studentsto describe any similaritiesthey notice between findingthe dimensions of the gardenand factoring a trinomial.
New
Teaching Tip Tell studentsthat the order in which theyrecord the factors does notmatter. So, (x � 4)(x � 2) isalso correct.
Factor x2 � 7x � 12.(x � 3)(x � 4)
Teaching Tip If students use thegraphing calculator to checktheir factoring, make sure theyclear all other functions fromthe Y= list, and clear all otherdrawings from the draw menu.
Factor x2 � 12x � 27.(x � 3)(x � 9)
PowerPoint®
Lesson 9-3 Factoring Trinomials: x2 � bx � c 491
SOLVE EQUATIONS BY FACTORING Some equations of the form x2 � bx � c � 0 can be solved by factoring and then using the Zero Product Property.
www.algebra1.com/extra_examples
b Is Positive and c Is NegativeFactor x2 � x � 12.
In this trinomial, b � 1 and c � �12. This means that m � n is positive and mn isnegative. So either m or n is negative, but not both. Therefore, make a list of thefactors of �12, where one factor of each pair is negative. Look for the pair offactors whose sum is 1.
Factors of �12 Sum of Factors
1, �12 �11�1, 12 11
2, �6 �4�2, 6 4
3, �4 �1�3, 4 1 The correct factors are �3 and �4.
x2 � x � 12 � (x � m)(x � n) Write the pattern.
� (x � 3)(x � 4) m � �3 and n � 4
Example 3Example 3
b Is Negative and c Is NegativeFactor x2 � 7x � 18.
Since b � �7 and c � �18, m � n is negative and mn is negative. So either m or nis negative, but not both.
Factors of �18 Sum of Factors
1, �18 �17�1, 18 17
2, �9 �7 The correct factors are 2 and �9.
x2 � 7x � 18 � (x � m)(x � n) Write the pattern.
� (x � 2)(x � 9) m � 2 and n � �9
Example 4Example 4
Solve an Equation by FactoringSolve x2 � 5x � 6. Check your solutions.
x2 � 5x � 6 Original equation
x2 � 5x � 6 � 0 Rewrite the equation so that one side equals 0.
(x � 1)(x � 6) � 0 Factor.
x � 1 � 0 or x � 6 � 0 Zero Product Property
x � 1 x � �6 Solve each equation.
The solution set is {1, �6}.
CHECK Substitute 1 and �6 for x in the original equation.
x2 � 5x � 6 x2 � 5x � 6
(1)2 � 5(1) � 6 (�6)2 � 5(�6) � 6
6 � 6 � 6 � 6 �
Example 5Example 5
Alternate MethodYou can use the oppositeof FOIL to factor trinomials.For instance, considerExample 3.
x2 � x � 12
(x � �)(x � �)
Try factor pairs of �12until the sum of theproducts of the Inner andOuter terms is x.
Study Tip← ←← ←
Lesson 9-3 Factoring Trinomials: x2 � bx � c 491
33
44
In-Class ExamplesIn-Class Examples
55
66
In-Class ExamplesIn-Class Examples
Teaching Tip Point out tostudents that whenever c isnegative, either m or n must benegative, because if a product isnegative, then one of the twofactors must be negative.
Factor x2 � 3x � 18.(x � 6)(x � 3)
Factor x2 � x � 20.(x � 5)(x � 4)
SOLVE EQUATIONS BY FACTORING
Teaching Tip Tell students to becareful when rewriting equationsso that one side equals zero.They must perform the sameoperation on both sides of theequation, and they must payattention to the signs in theresulting equation.
Solve x2 � 2x � 15. Checkyour solutions. {�5, 3}
ARCHITECTURE Marion has asmall art studio in her back-yard. She wants to build anew studio that has threetimes the area of the oldstudio by increasing the lengthand width by the sameamount. What are thedimensions of the new studio?
The dimensions of the new studioshould be 18 ft by 20 ft.
Study NotebookStudy NotebookHave students—• include explanations on how to
factor trinomials.• include any other item(s) that they
find helpful in mastering the skillsin this lesson.
FIND THE ERRORAsk students to
recall how manyanswers they usually get when
solving a trinomial equation.
Guided Practice
492 Chapter 9 Factoring
Solve a Real-World Problem by FactoringYEARBOOK DESIGN A sponsor for the school yearbook has asked that the length and width of a photo in their ad be increased by the same amount in order to double the area of the photo. If the photo was originally 12 centimeters wide by 8 centimeters long, what should the new dimensions of the enlarged photo be?
Explore Begin by making a diagram like the one shown above, labeling theappropriate dimensions.
Plan Let x � the amount added to each dimension of the photo.
The new length times the new width equals the new area.
Examine The solution set is {�24, 4}. Only 4 is a valid solution, since dimensionscannot be negative. Thus, the new length of the photo should be 4 � 12or 16 centimeters, and the new width should be 4 � 8 or 12 centimeters.
� �� �� �
12
8
x
x
Example 6Example 6
GUIDED PRACTICE KEYExercises Examples
4–9 1–410–15 5
16 6
Concept Check
2. Sample answer: x2 � 14x � 40 � 0;{4, 10}
1. Explain why, when factoring x2 � 6x � 9, it is not necessary to check the sum ofthe factor pairs �1 and �9 or �3 and �3. See margin.
2. OPEN ENDED Give an example of an equation that can be solved using thefactoring techniques presented in this lesson. Then, solve your equation.
3. FIND THE ERROR Peter and Aleta are solving x2 � 2x � 15.
Who is correct? Explain your reasoning. Aleta; to use the Zero Product Property, one side of the equation must equal zero.
1. In this trinomial, b � 6 and c � 9.This means that m � n is positiveand mn is positive. Only twopositive numbers have both apositive sum and product.Therefore, negative factors of 9need not be considered.
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
Less
on
9-3
Factor x2 � bx � c To factor a trinomial of the form x2 � bx � c, find two integers,m and n, whose sum is equal to b and whose product is equal to c.
Factoring x2 � bx � c x2 � bx � c � (x � m)(x � n), where m � n � b and mn � c.
Factor each trinomial.
a. x2 � 7x � 10In this trinomial, b � 7 and c � 10.
Since 2 � 5 � 7 and 2 � 5 � 10, let m � 2and n � 5.x2 � 7x � 10 � (x � 5)(x � 2)
b. x2 � 8x � 7In this trinomial, b � �8 and c � 7.Notice that m � n is negative and mn ispositive, so m and n are both negative.Since �7 � (�1) � �8 and (�7)(�1) � 7,m � �7 and n � �1.x2 � 8x � 7 � (x � 7)(x � 1)
Factors of 10 Sum of Factors
1, 10 11
2, 5 7
Factor x2 � 6x � 16.In this trinomial, b � 6 and c � �16. Thismeans m � n is positive and mn is negative.Make a list of the factors of �16, where onefactor of each pair is positive.
Therefore, m � �2 and n � 8.x2 � 6x � 16 � (x � 2)(x � 8)
31. Find all values of k so that the trinomial x2 � kx � 35 can be factored using integers.�34, �2, 2, 34
CONSTRUCTION For Exercises 32 and 33, use the following information.A construction company is planning to pour concrete for a driveway. The length of thedriveway is 16 feet longer than its width w.
32. Write an expression for the area of the driveway. w (w � 16) ft2
33. Find the dimensions of the driveway if it has an area of 260 square feet. 10 ft by 26 ft
WEB DESIGN For Exercises 34 and 35, use the following information.Janeel has a 10-inch by 12-inch photograph. She wants to scan the photograph, then reducethe result by the same amount in each dimension to post on her Web site. Janeel wants thearea of the image to be one eighth that of the original photograph.
34. Write an equation to represent the area of the reduced image.(10 � x)(12 � x) � 15, or x2 � 22x � 105 � 0
35. Find the dimensions of the reduced image. 3 in. by 5 in.
Practice (Average)
Factoring Trinomials: x2 � bx � c
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3Skills Practice, p. 537 and Practice, p. 538 (shown)
Reading to Learn Mathematics
Factoring Trinomials: x2 � bx � c
NAME ______________________________________________ DATE ____________ PERIOD _____
9-39-3
Less
on
9-3
Pre-Activity How can factoring be used to find the dimensions of a garden?
Read the introduction to Lesson 9-3 at the top of page 489 in your textbook.• Why do you need to find two numbers whose product is 54?
The problem asks you to find the length and width of thegarden. You know the area is 54 ft2. Since you multiplylength times width to find area, you need to find twonumbers whose product is 54.
• Why is the sum of these two numbers half the perimeter or 15?You add to find the perimeter of the garden. Since you usethe length twice and the width twice to find the perimeter,the sum of the length and the width is half the perimeter or 15.
Reading the LessonTell what sum and product you want m and n to have to use the pattern (x � m)(x � n) to factor the given trinomial.
1. x2 � 10x � 24 sum: product:
2. x2 � 12x � 20 sum: product:
3. x2 � 4x � 21 sum: product:
4. x2 � 6x � 16 sum: product:
5. To factor x2 � 18x � 32, you can look for numbers with a product of 32 and a sum of �18.Explain why the numbers in the pair you are looking for must both be negative.
To have a product of positive 32, the numbers must both be positive orboth be negative. If both were positive, their sum would be positiveinstead of negative.
Helping You Remember
6. If you are using the pattern (x � m)(x � n) to factor a trinomial of the form x2 � bx � c,how can you use your knowledge of multiplying integers to help you remember whetherm and n are positive or negative factors?
Sample answer: Like signs multiplied together result in positive numbersand unlike signs result in negative numbers. So, if c is positive, thefactors m and n both have the same sign as b. If c is negative, one of thefactors is positive and the other is negative.
�166
�21�4
20�12
2410
Reading to Learn Mathematics, p. 539
Puzzling PrimesA prime number has only two factors, itself and 1. The number 6 isnot prime because it has 2 and 3 as factors; 5 and 7 are prime. Thenumber 1 is not considered to be prime.
1. Use a calculator to help you find the 25 prime numbers less than 100.
Prime numbers have interested mathematicians for centuries. Theyhave tried to find expressions that will give all the prime numbers, oronly prime numbers. In the 1700s, Euler discovered that the trinomialx2 � x � 41 will yield prime numbers for values of x from 0 through 39.
2. Find the prime numbers generated by Euler’s formula for x from 0 through 7.
41, 43, 47, 53, 61, 71, 83, 97
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
54. SUPREME COURT When the Justices of the Supreme Court assemble to go onthe Bench each day, each Justice shakes hands with each of the other Justices for a total of 36 handshakes. The total number of handshakes h possible for
n people is given by h � �n2
2� n�. Write and solve an equation to determine the
number of Justices on the Supreme Court.
55. NUMBER THEORY Find two consecutive even integers whose product is 168.�14 and �12 or 12 and 14
56. GEOMETRY The triangle has an area of 40 square centimeters. Find the height h of the triangle. 5 cm
CRITICAL THINKING Find all values of k so that each trinomial can be factoredusing integers.
Speaking Ask volunteers tobrainstorm a mnemonic devicethat will help them rememberhow to factor trinomials withdifferent positive and negativevalues of b and c. Then writeexample trinomials on thechalkboard and have studentsfactor them using the mnemonicdevices as a guide.
Getting Ready for Lesson 9-4PREREQUISITE SKILL Studentswill learn to factor additionaltypes of trinomials in Lesson 9-4using factoring by grouping. UseExercises 78–83 to determineyour students’ familiarity withfactoring by grouping.
Assessment Options
Quiz (Lesson 9-3) is available onp. 573 of the Chapter 9 ResourceMasters.Mid-Chapter Test (Lessons 9-1through 9-3) is available on p. 575 of the Chapter 9 ResourceMasters.
Answer
63. Answers should include thefollowing.
• You would use a guess-and-check process, listing thefactors of 54, checking to seewhich pairs added to 15.
• To factor a trinomial of the formx2 � ax � c, you also use aguess-and-check process, listthe factors of c, and check tosee which ones add to a.
Maintain Your SkillsMaintain Your Skills
Solve each equation. Check your solutions. (Lesson 9-2)
70. (x � 3)(2x � 5) � 0 71. b(7b � 4) � 0 �0, �47
�� 72. 5y2 � �9y ���95
�, 0�Find the GCF of each set of monomials. (Lesson 9-1)
INTERNET For Exercises 76 and 77, use the graph at the right.(Lessons 3-7 and 8-3)
76. Find the percent increase in the number of domain registrations from 1997 to 2000. 1731%
77. Use your answer from Exercise 76 to verify the claim that registrations grew more than 18-fold from 1997 to 2000 iscorrect. 1(1.54) � 17.31(1.54) �(1 � 17.31)(1.54) or 18.31(1.54)
PREREQUISITE SKILL Factor each polynomial.(To review factoring by grouping, see Lesson 9-2.)
Number of domainregistrations climbsDuring the past four years,.com, .net and .org domainregistrations have grownmore than 18-fold:
28.2million
1997 1998 1999 2000
1.54million
3.36million
9million
494 Chapter 9 Factoring
4 Assess4 Assess
Online Lesson Plans
USA TODAY Education’s Online site offers resources andinteractive features connected to each day’s newspaper.Experience TODAY, USA TODAY’s daily lesson plan, isavailable on the site and delivered daily to subscribers.This plan provides instruction for integrating USA TODAYgraphics and key editorial features into your mathematicsclassroom. Log on to www.education.usatoday.com.
Mathematical Background notesare available for this lesson on p. 472D.
can algebra tiles be usedto factor 2x2 � 7x � 6?
Ask students:• When you form a rectangle
with algebra tiles to model atrinomial such as the one given,how do you know the factors?The length of the rectangle is onefactor, and the height is the otherfactor.
• How is the trinomial 2x2 � 7x � 6 different fromthose that you learned how tofactor in Lesson 9-3? The x2 termis multiplied by a constant (2).
• What would the rectangleformed by the given algebratiles look like? What are thefactors of the trinomial?
(2x � 3)(x � 2)
x2
1 1 1
x2 x x x
xx1 1 1xx
FACTOR ax2 � bx � c For trinomials of the form x2 � bx � c, the coefficientof x2 is 1. To factor trinomials of this form, you find the factors of c whose sum is b.We can modify this approach to factor trinomials whose leading coefficient is not 1.
F O I L(2x � 5)(3x � 1) � 6x2 � 2x � 15x � 5 Use the FOIL method.
2 � 15 � 17 and 2 � 15 � 6 � 5 m � n � b and mn � ac
You can use this pattern and the method of factoring by grouping to factor 6x2 � 17x � 5. Find two numbers, m and n, whose product is 6 � 5 or 30 and whosesum is 17.
Factors of 30 Sum of Factors
1, 30 312, 15 17 The correct factors are 2 and 15.
� (6x2 � 2x) � (15x � 5) Group terms with common factors.
� 2x(3x � 1) � 5(3x � 1) Factor the GCF from each grouping.
� (3x � 1)(2x � 5) 3x � 1 is the common factor.
Therefore, 6x2 � 17x � 5 � (3x � 1)(2x � 5).
Factoring Trinomials: ax2 � bx � c
Lesson 9-4 Factoring Trinomials: ax2 � bx � c 495
Vocabulary• prime polynomial
• Factor trinomials of the form ax2 � bx � c.
• Solve equations of the form ax2 � bx � c � 0.
The factors of 2x2 � 7x � 6 are the dimensions of the rectangle formed by thealgebra tiles shown below.
The process you use to form the rectangle is the same mental process you canuse to factor this trinomial algebraically.
x 21
1
1
1
1
1
xx 2 x x x x x x
Look BackTo review factoring bygrouping, see Lesson 9-2.
Study Tip
can algebra tiles be used to factor 2x2 � 7x � 6?can algebra tiles be used to factor 2x2 � 7x � 6?
Lesson x-x Lesson Title 495
Chapter 9 Resource Masters• Study Guide and Intervention, pp. 541–542• Skills Practice, p. 543• Practice, p. 544• Reading to Learn Mathematics, p. 545• Enrichment, p. 546
Graphing Calculator and Spreadsheet Masters, p. 39
In-Class ExamplesIn-Class ExamplesFACTOR ax2 � bx � c
Teaching Tip In Example 1b ofthe Student Edition, mn is alarge number, which has quite afew factors. Have students startlisting the factors that can easilybe determined by mental math.More than likely, the factors thatequal m � n can be found thisway, without too muchcalculation.
a. Factor 5x2 � 27x � 10.(5x � 2)(x � 5)
b. Factor 24x2 � 22x � 3.(4x � 3)(6x � 1)
Teaching Tip Remind studentsnot to forget to record the com-mon factor that they factoredout of the trinomial when theyrecord the other two factors.
Factor 4x2 � 24x � 32.4(x � 2)(x � 4)
Sometimes the terms of a trinomial will contain a common factor. In these cases,first use the Distributive Property to factor out the common factor. Then factor thetrinomial.
496 Chapter 9 Factoring
Factor ax2 � bx � ca. Factor 7x2 � 22x � 3.
In this trinomial, a � 7, b � 22 and c � 3. You need to find two numbers whosesum is 22 and whose product is 7 • 3 or 21. Make an organized list of the factorsof 21 and look for the pair of factors whose sum is 22.
� (7x2 � 1x) � (21x � 3) Group terms with common factors.
� x(7x � 1) � 3(7x � 1) Factor the GCF from each grouping.
� (7x � 1)(x � 3) Distributive Property
CHECK You can check this result by multiplying the two factors.
F O I L(7x � 1)(x � 3) � 7x2 � 21x � x � 3 FOIL method
� 7x2 � 22x � 3 � Simplify.
b. Factor 10x2 � 43x � 28.In this trinomial, a � 10, b � �43 and c � 28. Since b is negative, m � n isnegative. Since c is positive, mn is positive. So m and n must both be negative.Therefore, make a list of the negative factors of 10 � 28 or 280, and look for thepair of factors whose sum is �43.
Factors of 280 Sum of Factors
�1, �280 �281�2, �140 �142�4, �70 �74�5, �56 �61�7, �40 �47�8, �35 �43 The correct factors are �8 and �35.
10x2 � 43x � 28
� 10x2 � mx � nx � 28 Write the pattern.
� 10x2 � (�8)x � (�35)x � 28 m � �8 and n � �35
� (10x2 � 8x) � (�35x � 28) Group terms with common factors.
� 2x(5x � 4) � 7(�5x � 4) Factor the GCF from each grouping.
Finding FactorsFactor pairs in anorganized list so you donot miss any possiblepairs of factors.
Study Tip
Factor When a, b, and c Have a Common FactorFactor 3x2 � 24x � 45.
Notice that the GCF of the terms 3x2, 24x, and 45 is 3. When the GCF of the termsof a trinomial is an integer other than 1, you should first factor out this GCF.
Interpersonal Place students in groups to factor polynomials such asthose in Examples 1 and 2. Have each group member find one or twofactors for mn, depending on the number of factors and number ofstudents in the group. By dividing the labor, students should be able toquickly find the factors for mn that sum to m � n. Once they find thefactors, have students complete the factoring as a group.
Differentiated Instruction
PowerPoint®
Lesson 9-4 Factoring Trinomials: ax2 � bx � c 497
A polynomial that cannot be written as a product of two polynomials withintegral coefficients is called a .prime polynomial
www.algebra1.com/extra_examples
FactoringCompletelyAlways check for a GCFfirst before trying to factora trinomial.
Study Tip
Now factor x2 � 8x � 15. Since the lead coefficient is 1, find two factors of 15whose sum is 8.
Factors of 15 Sum of Factors
1, 15 163, 5 8 The correct factors are 2 and 15.
So, x2 � 8x � 15 � (x � 3)(x � 5). Thus, the complete factorization of 3x2 � 24x � 45 is 3(x � 3)(x � 5).
Determine Whether a Polynomial Is PrimeFactor 2x2 � 5x � 2.
In this trinomial, a � 2, b � 5 and c � �2. Since b is positive, m � n is positive.Since c is negative, mn is negative. So either m or n is negative, but not both.Therefore, make a list of the factors of 2 � �2 or �4, where one factor in each pairis negative. Look for a pair of factors whose sum is 5.
Factors of �4 Sum of Factors
1, �4 �3�1, 4 3�2, 2 0
There are no factors whose sum is 5. Therefore, 2x2 � 5x � 2 cannot be factoredusing integers. Thus, 2x2 � 5x � 2 is a prime polynomial.
Example 3Example 3
Solve Equations by FactoringSolve 8a2 � 9a � 5 � 4 � 3a. Check your solutions.
8a2 � 9a � 5 � 4 � 3a Original equation
8a2 � 6a � 9 � 0 Rewrite so that one side equals 0.
(4a � 3)(2a � 3) � 0 Factor the left side.
4a � 3 � 0 or 2a � 3 � 0 Zero Product Property
4a � �3 2a � 3 Solve each equation.
a � ��34
� a � �32
�
The solution set is ���34
�, �32
��.
CHECK Check each solution in the original equation.
8a2 � 9a � 5 � 4 � 3a 8a2 � 9a � 5 � 4 � 3a
8��34
�2� 9��
34
� � 5 � 4 � 3��34
� 8�32
�2� 9�
32
� � 5 � 4 � 3�32
��92
� � �247� � 5 � 4 � �
94
� 18 � �227� � 5 � 4 � �
92
�
�245� � �
245� � ��
12
� � ��12
� �
Example 4Example 4
SOLVE EQUATIONS BY FACTORING Some equations of the form ax2 � bx � c � 0 can be solved by factoring and then using the Zero Product Property.
Lesson 9-4 Factoring Trinomials: ax2 � bx � c 497
Dividing Students may assume that because they cannot divide eachside of x(x � 2) � 0 by x, they may never divide both sides of anequation set equal to zero by anything. Make sure they realize that theycan divide each side by a known number, as in Example 5, but theycannot divide each side by a variable that may equal zero.
Unlocking Misconceptions
33
In-Class ExampleIn-Class Example
44
55
In-Class ExamplesIn-Class Examples
Teaching Tip Make sure stu-dents list all possible factors ofmn, including both positive andnegative factors, before theydecide the polynomial is prime.
Factor 3x2 � 7x � 5. prime
SOLVE EQUATIONS BYFACTORING
Teaching Tip Remind studentsthat when a polynomial has twofactors, there are two solutions.Check each solution by substitu-ting it into the original equation.
Solve 18b2 � 19b � 8 � 3b2 � 5b.
�� , �MODEL ROCKETS Ms.Nguyen’s science class builtan air-launched model rocketfor a competition. When theytest-launched their rocketoutside the classroom, therocket landed in a nearbytree. If the launch pad was 2 feet above the ground, theinitial velocity of the rocketwas 64 feet per second, andthe rocket landed 30 feet abovethe ground, how long was therocket in flight? Use theequation h � �16t2 � vt � s.3.5 seconds
Study NotebookStudy NotebookHave students—• add the definitions/examples of
the vocabulary terms to theirVocabulary Builder worksheets forChapter 9.
• include explanations on how tofactor trinomials of the form ax2 � bx � c.
• include any other item(s) that theyfind helpful in mastering the skillsin this lesson.
FIND THE ERRORStudents should
first look at the num-bers for which Dasan and
Craig are finding factors. Thisclue should immediately tellthem which student is correct.Then, challenge students to findanother error in Dasan’s work.Students should notice that hefactored a 2 out of 2x2 � 11x � 18,which is not possible.
A model for the vertical motion of a projected object is given by the equation h � �16t2 � vt � s, where h is the height in feet, t is the time in seconds, v is the initialupward velocity in feet per second, and s is the starting height of the object in feet.
498 Chapter 9 Factoring
Solve Real-World Problems by FactoringPEP RALLY At a pep rally, small foam footballs are launched by cheerleaders using a sling-shot.How long is a football in the air if a student inthe stands catches it on its way down 26 feetabove the gym floor?
Use the model for vertical motion.
h � �16t2 � vt � s Vertical motion model
26 � �16t2 � 42t � 6 h � 26, v � 42, s � 6
0 � �16t2 � 42t � 20 Subtract 26 from each side.
0 � �2(8t2 � 21t � 10) Factor out �2.
0 � 8t2 � 21t � 10 Divide each side by �2.
0 � (8t � 5)(t � 2) Factor 8t2 � 21t � 10.
8t � 5 � 0 or t � 2 � 0 Zero Product Property
8t � 5 t � 2 Solve each equation.
t � �58
�
The solutions are �58
� second and 2 seconds. The first time represents how long it
takes the football to reach a height of 26 feet on its way up. The later time representshow long it takes the ball to reach a height of 26 feet again on its way down. Thus,the football will be in the air for 2 seconds before the student catches it.
6 ft
Height ofrelease
Height ofreception
26 ft
t � 0v � 42 ft/s
Example 5Example 5
1. Explain how to determine which values should be chosen for m and n whenfactoring a polynomial of the form ax2 � bx � c.
2. OPEN ENDED Write a trinomial that can be factored using a pair of numberswhose sum is 9 and whose product is 14. Sample answer: 2x2 � 9x � 7
3. FIND THE ERROR Dasan and Craig are factoring 2x2 � 11x � 18.
Who is correct? Explain your reasoning. Craig; see margin for explanation.
Factor each trinomial, if possible. If the trinomial cannot be factored usingintegers, write prime. 4. (3a � 2)(a � 2) 6. 2(p � 3)(p � 4)4. 3a2 � 8a � 4 5. 2a2 � 11a � 7 prime 6. 2p2 � 14p � 24
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
Less
on
9-4
Factor ax2 � bx � c To factor a trinomial of the form ax2 � bx � c, find two integers,m and n whose product is equal to ac and whose sum is equal to b. If there are no integersthat satisfy these requirements, the polynomial is called a prime polynomial.
Factor 2x2 � 15x � 18.In this example, a � 2, b � 15, and c � 18.You need to find two numbers whose sum is15 and whose product is 2 � 18 or 36. Makea list of the factors of 36 and look for thepair of factors whose sum is 15.
Use the pattern ax2 � mx � nx � c, with a � 2, m � 3, n � 12, and c � 18.
�� , � � , 2� �� , �31. DIVING Lauren dove into a swimming pool from a 15-foot-high diving board with an
initial upward velocity of 8 feet per second. Find the time t in seconds it took Lauren toenter the water. Use the model for vertical motion given by the equation h � �16t2 � vt � s, where h is height in feet, t is time in seconds, v is the initial upwardvelocity in feet per second, and s is the initial height in feet. (Hint: Let h � 0 representthe surface of the pool.) 1.25 s
32. BASEBALL Brad tossed a baseball in the air from a height of 6 feet with an initial upwardvelocity of 14 feet per second. Enrique caught the ball on its way down at a point 4 feetabove the ground. How long was the ball in the air before Enrique caught it? Use themodel of vertical motion from Exercise 31. 1 s
1�6
4�3
3�4
1�4
1�3
4�3
5�4
5�3
1�2
2�3
5�2
2�5
3�10
4�3
1�2
3�4
1�2
2�5
1�3
8�3
Practice (Average)
Factoring Trinomials: ax2 � bx � c
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4Skills Practice, p. 543 and Practice, p. 544 (shown)
Reading to Learn Mathematics
Factoring Trinomials: ax2 � bx � c
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4
Less
on
9-4
Pre-Activity How can algebra tiles be used to factor 2x2 � 7x � 6?
Read the introduction to Lesson 9-4 at the top of page 495 in your textbook.
• When you form the algebra tiles into a rectangle, what is the first step?
Place the two x2 tiles on the product mat and arrange the six 1 tiles into a rectangular array.
• What is the second step?
Arrange the seven x tiles to complete the rectangle.
Reading the Lesson
1. Suppose you want to factor the trinomial 3x2 � 14x � 8.
a. What is the first step?
Find integers with a product of 24 and a sum of 14. The integers are 2and 12.
b. What is the second step?
Rewrite the polynomial by breaking the middle term into two addendsthat use 2 and 12 as coefficients. You can use 3x2 � 14x � 8 � 3x2 � 2x � 12x � 8.
c. Provide an explanation for the next two steps.
(3x2 � 2x) � (12x � 8) Group terms with common factors.x(3x � 2) � 4(3x � 2) Factor the GCF from each grouping.
d. Use the Distributive Property to rewrite the last expression in part c. You get
( � )(3x � 2).
2. Explain how you know that the trinomial 2x2 � 7x � 4 is a prime polynomial.
To factor 2x2 � 7x � 4, you would need two negative integers whoseproduct is 8 and whose sum is �7. There are no such negative integers.Therefore, 2x2 � 7x � 4 is prime.
Helping You Remember
3. What are steps you could use to remember how to find the factors of a trinomial writtenin the form of ax2 � bx � c?
Sample answer: Look for two integers with a product equal to ac and asum of b. Replace the middle term with a sum of x terms that have thosetwo integers as coefficients. Then factor by grouping.
4x
Reading to Learn Mathematics, p. 545
Area Models for Quadratic TrinomialsAfter you have factored a quadratic trinomial, you can use the factors to draw geometric models of the trinomial.
x2 � 5x � 6 � (x � 1)(x � 6)
To draw a rectangular model, the value 2 was used for x so that the shorter side would have a length of 1. Then the drawing was done in centimeters. So, the area of the rectangle is x2 � 5x � 6.
To draw a right triangle model, recall that the area of a triangle is one-half the base times the height. So, one of the sides must be twice as long as the shorter side of therectangular model.
x2 � 5x � 6 � (x � 1)(x � 6)
� (2x � 2)(x � 6)
The area of the right triangle is also x2 � 5x � 6.
1�2
x � 6
2x � 2
x � 6
x � 1
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-49-4Enrichment, p. 546
Lesson 9-4 Factoring Trinomials: ax2 � bx � c 499
10. ��3, ��23
��Application
www.algebra1.com/self_check_quiz
Practice and ApplyPractice and Apply
indicates increased difficulty�
Cliff DivingIn Acapulco, Mexico, diversleap from La Quebrada, the “Break in the Rocks,”diving headfirst into thePacific Ocean 105 feetbelow. Source: acapulco-travel.
web.com.mx
Factor each trinomial, if possible. If the trinomial cannot be factored usingintegers, write prime. 14–31. See margin.14. 2x2 � 7x � 5 15. 3x2 � 5x � 2 16. 6p2 � 5p � 6
GEOMETRY For Exercises 49 and 50, use the followinginformation.A rectangle with an area of 35 square inches is formed by cutting off strips of equal width from a rectangular piece of paper.
49. Find the width of each strip. 1 in.50. Find the dimensions of the new rectangle. 5 in. by 7 in.
51. CLIFF DIVING Suppose a diver leaps from the edge of a cliff 80 feet above theocean with an initial upward velocity of 8 feet per second. How long it will takethe diver to enter the water below? 2.5 s
gymnast making a vaultleaves the horse, her feet are8 feet above the groundtraveling with an initialupward velocity of 8 feet persecond. Use the model forvertical motion to find thetime t in seconds it takes forthe gymnast’s feet to reach themat. (Hint: Let h � 0, theheight of the mat.) 1 s
Writing Place students in pairs.Have each student pick a problemthat involves solving an equationby factoring and intentionallymake a mistake while solvingthe problem. Then have studentsexchange problems. Studentsshould write a description of whatis incorrect, and how to solve theproblem correctly. Have studentsshare their explanations andcorrect solutions with the class.
Getting Ready for Lesson 9-5PREREQUISITE SKILL Students willlearn to factor binomials that aredifferences of squares in Lesson9-5. Factoring differences ofsquares requires students to beable to quickly find the principalsquare roots of perfect squares.Use Exercises 63–70 to determineyour students’ familiarity withfinding square roots.
Assessment Options
Practice Quiz 2 The quiz pro-vides students with a brief reviewof the concepts and skills inLessons 9-3 and 9-4. Lessonnumbers are given to the right ofthe exercises or instruction linesso students can review conceptsnot yet mastered.
Answer53. You can use algebra tiles to factor
2x2 � 7x � 6 by finding the dimen-sions of the rectangle that is formedby the tiles for 2x2 � 7x � 6. An-swers should include the following.• 2x � 3 by x � 2• With algebra tiles, you can try
various ways to make a rectanglewith the necessary tiles. Onceyou make the rectangle, how-ever, the dimensions of therectangle are the factors of thepolynomial. In a way, you have togo through the guess-and-checkprocess whether you are factor-ing algebraically or geometri-cally (using algebra tiles.)
500 Chapter 9 Factoring
StandardizedTest Practice
Maintain Your SkillsMaintain Your Skills
Practice Quiz 2Practice Quiz 2
Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lessons 9-3 and 9-4)
62. BUSINESS Jake’s Garage charges $83 for a two-hour repair job and $185 for a five-hour repair job. Write a linear equation that Jake can use to bill customersfor repair jobs of any length of time. (Lesson 5-3) y � 34x � 15
PREREQUISITE SKILL Find the principal square root of each number.(To review square roots, see Lesson 2-7.)
52. CLIMBING Damaris launches a grappling hook from a height of 6 feet with aninitial upward velocity of 56 feet per second. The hook just misses the stoneledge of a building she wants to scale. As it falls, the hook anchors on the ledge,which is 30 feet above the ground. How long was the hook in the air? 3 s
53. Answer the question that was posed at the beginning of the lesson. See margin.
How can algebra tiles be used to factor 2x2 � 7x � 6?
Include the following in your answer:
• the dimensions of the rectangle formed, and
• an explanation, using words and drawings, of how this geometric guess-and-check process of factoring is similar to the algebraic processdescribed on page 495.
54. What are the solutions of 2p2 � p � 3 � 0? D��
23
� and 1 �23
� and �1 ��32
� and 1 �32
� and �1
55. Suppose a person standing atop a building 398 feet tall throws a ball upward. Ifthe person releases the ball 4 feet above the top of the building, the ball’s heighth, in feet, after t seconds is given by the equation h � �16t2 � 48t � 402. Afterhow many seconds will the ball be 338 feet from the ground? B
3.5 4 4.5 5DCBA
DCBA
WRITING IN MATH
500 Chapter 9 Factoring
4 Assess4 Assess
Guess (2x � 1)(x � 3) incorrectbecause 8 x tiles are needed tocomplete the rectangle
x 2 x x
x 2 x x
xxx
1 1
1 11 1
x 2 x x x
x 2 x x x
x 1 1 1
1 1 1
5-Minute CheckTransparency 9-5 Use as a
quiz or review of Lesson 9-4.
Mathematical Background notesare available for this lesson on p. 472D.
Building on PriorKnowledge
In Chapter 8, students learnedabout a special binomial productcalled a difference of squares.Specifically, a difference of squaresis the product of two binomials ofthe form (a � b)(a � b), and theproduct is of the form a2 � b2. Inthis lesson, students will learn tofactor differences of squares, anduse the factoring of differences ofsquares to solve equations.
can you determine abasketball player’shang time?
Ask students:• What is a perfect square?
A perfect square is a rationalnumber whose square root is arational number.
• Is 4t2 a perfect square? If so,what is its principal squareroot? Yes, the principal squareroot is 2t.
• If a basketball player can jump4 feet, what would be her hangtime? 1 second
Factoring Differences of Squares
• Factor binomials that are the differences of squares.
• Solve equations involving the differences of squares.
A basketball player’s hang time is the length of time he is in the air after jumping. Given themaximum height h a player can jump, you candetermine his hang time t in seconds by solving 4t2 � h � 0. If h is a perfect square, this equation can be solved by factoring using the pattern for the difference of squares.
Use a straightedge to drawtwo squares similar to thoseshown below. Choose anymeasures for a and b.
Notice that the area of thelarge square is a2, and thearea of the small square is b2.
Cut the irregular region intotwo congruent pieces asshown below.
Cut the small square fromthe large square.
The area of the remainingirregular region is a2 � b2.
Rearrange the two congruent regions to form a rectangle with length a � b and width a � b.
a
a � b
a
b
b
Region 2
Region 1a � b
a � b
a
a � b
a � b
a
b b
b b
a
a � b
a � b
a
b
b
Region 2
Reg
ion
1
a
a
b
b
Look BackTo review the
product of a sum
and a difference,
see Lesson 8-8.
Study Tip
Make a Conjecture1. Write an expression representing the area of the rectangle. (a � b)(a � b)2. Explain why a2 � b2 = (a � b)(a � b). Since a2 � b2 and (a � b)(a � b)
describe the same area, a2 � b2 � (a � b)(a � b).
FACTOR a2 � b2 A geometric model can be used to factor the difference of squares.
Lesson 9-5 Factoring Differences of Squares 501
can you determine a basketball player’s hang time?can you determine a basketball player’s hang time?
Difference of Squares
Lesson x-x Lesson Title 501
Chapter 9 Resource Masters• Study Guide and Intervention, pp. 547–548• Skills Practice, p. 549• Practice, p. 550• Reading to Learn Mathematics, p. 551• Enrichment, p. 552• Assessment, p. 574Prerequisite Skills Workbook, pp. 13–14
Graphing Calculator and Spreadsheet Masters, p. 40
Parent and Student Study GuideWorkbook, p. 72
School-to-Career Masters, p. 18Science and Mathematics Lab Manual,
Teaching Tip Students maycheck their factoring bymultiplying the factors using theFOIL method. The first-degreeterm will always drop out whenthe product is a difference ofsquares.
Factor each binomial.
a. m2 � 64 (m � 8)(m � 8)
b. 16y2 � 81z2 (4y � 9z)(4y � 9z)
Factor 3b3 � 27b.3b (b � 3)(b � 3)
Teaching Tip Students shouldnotice that when the differenceof squares factoring techniquehas been applied once, one ofthe factors should be prime.
� (6x � 7y) (6x � 7y) Factor the difference of squares.
Example 1Example 1
CommonMisconceptionRemember that the sumof two squares, like x2 � 9, is not factorableusing the difference ofsquares pattern. x2 � 9is a prime polynomial.
Study Tip
Difference of Squares
Factor Out a Common FactorFactor 48a3 � 12a.
48a3 � 12a � 12a(4a2 � 1) The GCF of 48a3 and �12a is 12a.
� 12a[(2a)2 � 12] 4a2 � 2a � 2a and 1 � 1 � 1
� 12a(2a � 1)(2a � 1) Factor the difference of squares.
Example 2Example 2
Apply a Factoring Technique More Than OnceFactor 2x4 � 162.
2x4 � 162 � 2(x4 � 81) The GCF of 2x4 and �162 is 2.
� 2[(x2)2 � 92] x4 � x2 � x2 and 81 � 9 � 9
� 2(x2 � 9)(x2 � 9) Factor the difference of squares.
� 2(x2 � 9)(x2 � 32) x2 � x � x and 9 � 3 � 3
� 2(x2 � 9)(x � 3)(x � 3) Factor the difference of squares.
Example 3Example 3
If the terms of a binomial have a common factor, the GCF should be factored outfirst before trying to apply any other factoring technique.
Occasionally, the difference of squares pattern needs to be applied more than onceto factor a polynomial completely.
Apply Several Different Factoring TechniquesFactor 5x3 � 15x2 � 5x � 15.
5x3 � 15x2 � 5x � 15 Original polynomial
� 5(x3 � 3x2 � x � 3) Factor out the GCF.
� 5[(x3 � x) � (3x2 � 3)] Group terms with common factors.
� 5[x(x2 � 1) � 3(x2 � 1)] Factor each grouping.
� 5(x2 � 1)(x � 3) x2 � 1 is the common factor.
� 5(x � 1)(x � 1)(x � 3) Factor the difference of squares, x2 � 1, into (x � 1)(x � 1).
Example 4Example 4
502 Chapter 9 Factoring
2 Teach2 Teach
Algebra Activity
Materials: straightedge, scissors• Using graph paper, students are more likely to draw straight squares, which will
make the final product appear more like a rectangle.• Make sure students label their figures as shown. Explain that the sides of the
original square have length of a, and when the b square is cut out, theremaining sides have lengths of a � b.
PowerPoint®
Lesson 9-5 Factoring Differences of Squares 503
SOLVE EQUATIONS BY FACTORING You can apply the Zero ProductProperty to an equation that is written as the product of any number of factors setequal to 0.
www.algebra1.com/extra_examples
Solve Equations by FactoringSolve each equation by factoring. Check your solutions.
Test-Taking TipLook to see if the area ofan oddly-shaped figure canbe found by subtractingthe areas of more familiarshapes, such as triangles,rectangles, or circles.
A corner is cut off a 2-inch by 2-inch square piece of paper. The cut is x inches from a corner as shown.
a. Write an equation in terms of x that represents thearea A of the paper after the corner is removed.
b. What value of x will result in an area that is �79
� the
area of the original square piece of paper? Showhow you arrived at your answer.
x
x
2 in.
2 in.
(continued on the next page)
Lesson 9-5 Factoring Differences of Squares 503
Example 6 Explain to students that when they completeextended response test items, it is very important to show allthe steps in working out the problem, rather than just writingthe answer. Often, full credit is not given for an extended
response item unless all of the steps are shown. Conversely, students may receive partialcredit for an incorrect answer, if their work shows only an arithmetic error, but the problemwas solved using the correct method.
55
66
In-Class ExamplesIn-Class Examples
SOLVE EQUATIONS BYFACTORING
Teaching Tip Students may notbe used to thinking of fractionsas perfect squares. Remind themthat if both the numerator anddenominator are perfect squares,then the fraction itself is aperfect square. Also, if studentsare uncomfortable findingsquare roots of fractions, pointout the Study Tip in the text.
Solve each equation byfactoring. Check yoursolutions.
a. q2 � � 0 � , � �b. 48y3 � 3y �� , �
EXTENDED RESPONSE Asquare with side length x iscut from the right triangleshown below.
a. Write an equation in terms ofx that represents the area A ofthe triangle after the corner isremoved. A � 64 � x2
b. What value of x will result in
a triangle that is of the area
of the original triangle? Showhow you arrived at youranswer. 4
Solve each equation by factoring. Check your solutions.
11. 4y2 � 25 ���52
�, �52
�� 12. 17 � 68k2 � 0 ���12
�, �12
��13. x2 � �
316� � 0 ���1
6�, �1
6�� 14. 121a � 49a3
���171�, 0, �1
71��
Jess ica
64x2 + 16y2
= 16(4x2 + y2 )
= 16(2x + y ) (2x – y )
Manuel
64x2 + 16y2
= 16(4x2 + y2)
Concept Check 1. Each term of thebinomial is a perfectsquare, and the binomial can be written as a differenceof terms.3. Yes; 3n2 � 48 �3(n2 � 16) �3(n � 4)(n � 4).
Guided Practice
504 Chapter 9 Factoring
About the Exercises…Organization by Objective• Factor a2 � b2: 16–33• Solve Equations by
Factoring: 34–45
Odd/Even AssignmentsExercises 16–45 are structuredso that students practice thesame concepts whether theyare assigned odd or evenproblems.
Advanced: 16–50 even, 51–64(optional: 65–70) Intrapersonal Consider having students complete the Check for
Understanding problems on one day, and then check their own work onthe next day, examining their work and answers. Letting their own worksit for a day often allows students to see mistakes or problems in theirwork that they otherwise might not have noticed. It may also givestudents more of a chance to ask for help on difficult problems.
Differentiated Instruction
Study Guide and Intervention
Factoring Differences of Squares
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
Less
on
9-5
Factor a2 � b2 The binomial expression a2 � b2 is called the difference of twosquares. The following pattern shows how to factor the difference of squares.
Difference of Squares a2 � b2 � (a � b)(a � b) � (a � b)(a � b).
Factor each binomial.
a. n2 � 64n2 � 64
� n2 � 82 Write in the form a2 � b2.
� (n � 8)(n � 8) Factor.
b. 4m2 � 81n2
4m2 � 81n2
� (2m)2 � (9n)2 Write in the form a2 � b2.
� (2m � 9n)(2m � 9n) Factor.
Factor each polynomial.
a. 50a2 � 7250a2 � 72
� 2(25a2 � 36) Find the GCF.
� 2[(5a)2 � 62)] 25a2 � 5a � 5a and 36 � 6 � 6
�2(5a � 6)(5a � 6) Factor the difference of squares.
b. 4x4 � 8x3 � 4x2 � 8x4x4 � 8x3 � 4x2 � 8x Original polynomial
� 4x(x3 � 2x2 � x � 2) Find the GCF.
� 4x[(x3 � 2x2) � (x � 2)] Group terms.
� 4x[x2(x � 2) � 1(x � 2)] Find the GCF.
� 4x[(x2 � 1)(x � 2)] Factor by grouping.
� 4x[(x � 1)(x � 1)(x � 2)] Factor the difference
of squares.
Example 1Example 1 Example 2Example 2
ExercisesExercises
Factor each polynomial if possible. If the polynomial cannot be factored,write prime.
{30} � , 0� � , 0�25. EROSION A rock breaks loose from a cliff and plunges toward the ground 400 feet
below. The distance d that the rock falls in t seconds is given by the equation d � 16t2.How long does it take the rock to hit the ground? 5 s
26. FORENSICS Mr. Cooper contested a speeding ticket given to him after he applied hisbrakes and skidded to a halt to avoid hitting another car. In traffic court, he argued thatthe length of the skid marks on the pavement, 150 feet, proved that he was driving underthe posted speed limit of 65 miles per hour. The ticket cited his speed at 70 miles per hour.
Use the formula s2 � d, where s is the speed of the car and d is the length of the skid
marks, to determine Mr. Cooper’s speed when he applied the brakes. Was Mr. Coopercorrect in claiming that he was not speeding when he applied the brakes? 60 mi/h; yes
1�24
7�5
4�3
1�36
4�7
8�11
4�9
16�49
64�121
5�7
3�8
9�2
Practice (Average)
Factoring Differences of Squares
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5Skills Practice, p. 549 and Practice, p. 550 (shown)
Reading to Learn Mathematics
Factoring Differences of Squares
NAME ______________________________________________ DATE ____________ PERIOD _____
9-59-5
Less
on
9-5
Pre-Activity How can you determine a basketball player’s hang time?
Read the introduction to Lesson 9-5 at the top of page 501 in your textbook.
Suppose a player can jump 2 feet. Can you use the pattern for thedifference of squares to solve the equation 4t2 � 2 � 0? Explain.
No; 2 is not a perfect square.
Reading the Lesson
1. Explain why each binomial is a difference of squares.
a. 4x2 � 25
4x2 is the square of 2x, 25 is the square of 5, and the operation sign isa minus sign.
b. 49a2 � 64b2
49a2 is the square of 7a, 64b2 is the square of 8b, and the operationsign is a minus sign.
2. Sometimes it is necessary to apply more than one technique when factoring, or to applythe same technique more than once.
a. What should you look for first when you are factoring a binomial?
Look for a factor or factors common to the terms of the binomial.
b. Explain what is done in each step to factor 4x4 � 64.
4x4 � 64
� 4(x4 � 16) Factor out the GCF.� 4[(x2)2 � 42] Write x4 � 16 in difference of squares form.� 4(x2 � 4)(x2 � 4) Factor the difference of squares.� 4(x2 � 4)(x2 � 22) Write x2 � 4 in difference of squares form.� 4(x2 � 4)(x � 2)(x � 2) Factor the difference of squares.
3. Suppose you are solving the equation 16x2 � 9 � 0 and rewrite it as (4x � 3)(4x � 3) � 0.What would be your next steps in solving the equation?
Set each factor equal to zero, then solve the resulting equations.
Helping You Remember
4. How can you remember whether a binomial can be factored as a difference of squares?
The operation sign must be a minus sign, and the expressions beforeand after the minus sign must be perfect squares.
Reading to Learn Mathematics, p. 551
Factoring Trinomials of Fourth DegreeSome trinomials of the form a4 � a2b2 � b4 can be written as the difference of two squares and then factored.
Factor 4x4 � 37x2y2 � 9y4.
Step 1 Find the square roots of the first and last terms.
�4x4� � 2x2 �9y4� � 3y2
Step 2 Find twice the product of the square roots.
2(2x2)(3y2) � 12x2y2
Step 3 Separate the middle term into two parts. One part is either your answer to Step 2 or its opposite. The other part should be the opposite of a perfect square.
�37x2y2 � �12x2y2 � 25x2y2
Step 4 Rewrite the trinomial as the difference of two squares and then factor.
AerodynamicsLift works on the principlethat as the speed of a gasincreases, the pressuredecreases. As the velocityof the air passing over acurved wing increases, thepressure above the wingdecreases, lift is created,and the wing rises.Source: www.gleim.com
�, 4�{�3, �2, 2}46. CRITICAL THINKING Show that a2 � b2 � (a � b)(a � b) algebraically.
(Hint: Rewrite a2 � b2 as a2 � 0ab � b2.) See margin.
47. BOATING The United States Coast Guard’s License Exam includes questionsdealing with the breaking strength of a line. The basic breaking strength b inpounds for a natural fiber line is determined by the formula 900c2 � b, where c is the circumference of the line in inches. What circumference of natural linewould have 3600 pounds of breaking strength? 2 in.
48. AERODYNAMICS The formula for the pressure difference P above and below
a wing is described by the formula P � �12
�dv12 � �
12
�dv22, where d is the density of
the air, v1 is the velocity of the air passing above, and v2 is the velocity of the airpassing below. Write this formula in factored form. P � �1
2�d(v1 � v2)(v1 � v2)
49. LAW ENFORCEMENT If a car skids on dry concrete, police can use the formula
�214�s2 � d to approximate the speed s of a vehicle in miles per hour given the
length d of the skid marks in feet. If the length of skid marks on dry concrete are54 feet long, how fast was the car traveling when the brakes were applied? 36 mph
50. PACKAGING The width of a box is 9 inches more than its length. The height of the box is 1 inch less than its length. Ifthe box has a volume of 72 cubic inches,what are the dimensions of the box?3 in. by 12 in. by 2 in.
Modeling Have students com-plete an area problem similar toExample 6 using grid paper. Havethem demonstrate using the gridthat the area and lengths theycome up with using factoring adifference of squares is correct.
Getting Ready for Lesson 9-6PREREQUISITE SKILL Studentswill learn to factor perfect squaretrinomials in Lesson 9-6. Whenthey learn how to factor perfectsquare trinomials, students willhave to know how to squarebinomials. Use Exercises 65–70 todetermine your students’familiarity with finding specialproducts, such as the square of abinomial.
Assessment Options
Quiz (Lessons 9-4 and 9-5) isavailable on p. 574 of the Chapter 9Resource Masters.
Answers
52. Answers should include thefollowing.
• 1 foot
• To find the hang time of a stu-dent athlete who attains amaximum height of 1 foot, solvethe equation 4t2 � 1 � 0. Youcan factor the left side usingthe difference of squarespattern since 4t2 is the squareof 2t and 1 is the square of 1.Thus the equation becomes (2t � 1)(2t � 1) � 0. Using theZero Product Property, eachfactor can be set equal to zero,resulting in two solutions,
t � � and t � . Since time
cannot be negative, the hang
time is second.1�2
1�2
1�2
Maintain Your SkillsMaintain Your Skills
51. CRITICAL THINKING The following statements appear to prove that 2 is equalto 1. Find the flaw in this “proof.”
Suppose a and b are real numbers such that a � b, a 0, b 0.
(1) a � b Given.
(2) a2 � ab Multiply each side by a.
(3) a2 � b2 � ab � b2 Subtract b2 from each side.
(4) (a � b)(a � b) � b(a � b) Factor.
(5) a � b � b Divide each side by a � b.
(6) a � a � a Substitution Property; a � b
(7) 2a � a Combine like terms.
(8) 2 � 1 Divide each side by a.
52. Answer the question that was posed at the beginning of the lesson. See margin.
How can you determine a basketball player’s hang time?
Include the following in your answer:
• a maximum height that is a perfect square and that would be considered a reasonable distance for a student athlete to jump, and
• a description of how to find the hang time for this maximum height.
53. What is the factored form of 25b2 � 1? A(5b � 1)(5b � 1) (5b � 1)(5b � 1)
(5b � 1)(5b � 1) (25b � 1)(b � 1)
54. GRID IN In the figure, the area between the two squares is 17 square inches. The sum of the perimeters of the twosquares is 68 inches. How many inches long is a side of thelarger square? 9 in.
DC
BA
WRITING IN MATH
506 Chapter 9 Factoring
51. The flaw is in line5. Since a � b, a � b � 0. Thereforedividing by a � b isdividing by zero,which is undefined.
61. STATISTICS Amy’s scores on the first three of four 100-point biology tests were 88, 90, and 91. To get a B� in the class, her average must be between 88 and 92,inclusive, on all tests. What score must she receive on the fourth test to get a B� in biology? (Lesson 6-4) between 83 and 99, inclusive
Solve each inequality, check your solution, and graph it on a number line.(Lesson 6-1) 62–64. See margin for graphs.62. 6 3d � 12 d � 6 63. �5 � 10r � 2 r � �
170� 64. 13x � 3 � 23 x 2
PREREQUISITE SKILL Find each product. (To review special products, see Lesson 8-8.)
Before using this page, ask stu-dents to think of some commonmathematical symbols. As stu-dents suggest what they thinkare some of the more commonsymbols, write the symbols onthe chalkboard. Some examplesmight be:
� the addition sign� the subtraction sign� the equals sign
Interpreting Concepts Writeanother mathematical sentenceon the chalkboard, similar to theone in the text. Have studentsinterpret the sentence and list theconcepts that they have to knowin order to interpret the sentence.
Learning New Concepts Pointout to students that by skimminga new lesson before actually read-ing it to learn the concepts, theywill get an idea about what is im-portant in the lesson. The head-ings, bold words, and examplesillustrate the important concepts.
Ask students to summarize whatthey have learned about readingand interpreting mathematicalconcepts in their study notebooks.
Investigating Slope-Intercept Form 507Reading Mathematics The Language of Mathematics 507
Mathematics is a language all its own. As with any language you learn, you mustread slowly and carefully, translating small portions of it at a time. Then you mustreread the entire passage to make complete sense of what you read.
In mathematics, concepts are often written in a compact form by using symbols.Break down the symbols and try to translate each piece before putting them backtogether. Read the following sentence.
a2 � 2ab � b2 � (a � b)2
The trinomial a squared plus twice the product of a and b plus b squared equals the square of the binomial a plus b.
Below is a list of the concepts involved in that single sentence.
• The letters a and b are variables and can be replaced by monomials like 2 or 3x or by
polynomials like x � 3.
• The square of the binomial a � b means (a � b)(a � b). So, a2 � 2ab � b2 can be
written as the product of two identical factors, a � b and a � b.
Now put these concepts together. The algebraic statement a2 � 2ab � b2 � (a � b)2
means that any trinomial that can be written in the form a2 � 2ab � b2 can befactored as the square of a binomial using the pattern (a � b)2.
When reading a lesson in your book, use these steps.
• Read the “What You’ll Learn” statements to understand what concepts are being presented.
• Skim to get a general idea of the content.
• Take note of any new terms in the lesson by looking for highlighted words.
• Go back and reread in order to understand all of the ideas presented.
• Study all of the examples.
• Pay special attention to the explanations for each step in each example.
• Read any study tips presented in the margins of the lesson.
Reading to Learn 2. GCF, perfect square trinomial; x2 � bx � c, ax2 � bx � cTurn to page 508 and skim Lesson 9-6.1. List three main ideas from Lesson 9-6. Use phrases instead of whole
sentences. See margin.
2. What factoring techniques should be tried when factoring a trinomial?
3. What should you always check for first when trying to factor any polynomial? a greatest common factor
4. Translate the symbolic representation of the Square Root Property presented on page 511 and explain why it can be applied to problems like (a � 4)2 � 49 in Example 4a. See margin.
The Language of Mathematics
Reading Mathematics The Language of Mathematics 507
1. (1) explains how to factor a perfectsquare trinomial; (2) summarizesmethods used to factor polynomials; (3) explains how to solve equationsinvolving perfect squares using theSquare Root Property
4. For any number n, where n is positive,the square of x equals n, then x equalsplus or minus the square root of n.This property can be applied to theequation (a � 4)2 � 49 since thevariable x � a � 4 and n � 49 in theequation x2 � n.
English LanguageLearners may benefit fromwriting key concepts from thisactivity in their Study Notebooksin their native language and thenin English.
ELL
Answers
5-Minute CheckTransparency 9-6 Use as a
quiz or review of Lesson 9-5.
Mathematical Background notesare available for this lesson on p. 472D.
can factoring be used todesign a pavilion?
Ask students:• What does the expression
8 � 2x represent? The expressionrepresents the side length of theentire square pavilion.
• What does 144 represent?144 is the area of the entire squarepavilion.
• How do you know the area ofthe pavilion is 144 square feet?The square mascot area has an areaof 82, or 64 ft2 because its sidelength is 8 ft. The problem statesthat the bricks will cover 80 ft2, sothe area of the entire structure is64 � 80, or 144 ft2.
• What feature of the pavilion tellsyou that 144 must be a perfectsquare? The building is square,so the area must be the square ofthe side length. Since the area is144, 8 � 2x � 12, and x � 2.
FACTOR PERFECT SQUARE TRINOMIALS Numbers like 144, 16, and 49are perfect squares, since each can be expressed as the square of an integer.
144 � 12 � 12 or 122 16 � 4 � 4 or 42 49 � 7 � 7 or 72
Products of the form (a � b)2 and (a � b)2, such as (8 � 2x)2, are also perfect squares.Recall that these are special products that follow specific patterns.
(a � b)2 � (a � b)(a � b) (a � b)2 � (a � b)(a � b)
� a2 � ab � ab � b2 � a2 � ab � ab � b2
� a2 � 2ab � b2 � a2 � 2ab � b2
These patterns can help you factor , trinomials that arethe square of a binomial.
perfect square trinomials
Vocabulary• perfect square
trinomials
Perfect Squares and Factoring
508 Chapter 9 Factoring
• Factor perfect square trinomials.
• Solve equations involving perfect squares.
The senior class has decided to build an outdoor pavilion. It will have an 8-foot by 8-foot portrayal ofthe school’s mascot in the center. The class is sellingbricks with students’ names on them to finance theproject. If they sell enough bricks to cover 80 squarefeet and want to arrange the bricks around the art,how wide should the border of bricks be? To solvethis problem, you would need to solve the equation(8 � 2x)2 � 144.
8 ft
8 ftxxx
x
x
can factoring be used to design a pavilion?
Look BackTo review the square of asum or difference, seeLesson 8-8.
Study Tip
For a trinomial to be factorable as a perfect square, three conditions must besatisfied as illustrated in the example below.
4x2 � 20x � 25
The middle term must be twicethe product of the square roots
Chapter 9 Resource Masters• Study Guide and Intervention, pp. 553–554• Skills Practice, p. 555• Practice, p. 556• Reading to Learn Mathematics, p. 557• Enrichment, p. 558• Assessment, p. 574
Factoring Perfect Square Trinomials• Words If a trinomial can be written in the form a2 � 2ab � b2 or a2 � 2ab � b2,
then it can be factored as (a � b)2 or as (a � b)2, respectively.
• Symbols a2 � 2ab � b2 � (a � b)2 and a2 � 2ab � b2 � (a � b)2
• Example 4x2 � 20x � 25 � (2x)2 � 2(2x)(5) � (5)2 or (2x � 5)2
In this chapter, you have learned to factor different types of polynomials. TheConcept Summary lists these methods and can help you decide when to use aspecific method.
Factoring Polynomials
Factor Perfect Square TrinomialsDetermine whether each trinomial is a perfect square trinomial. If so, factor it.
a. 16x2 � 32x � 64
Is the first term a perfect square? Yes, 16x2 � (4x)2.Is the last term a perfect square? Yes, 64 � 82.Is the middle term equal to 2(4x)(8)? No, 32x 2(4x)(8).
16x2 � 32x � 64 is not a perfect square trinomial.
b. 9y2 � 12y � 4
Is the first term a perfect square? Yes, 9y2 � (3y)2.Is the last term a perfect square? Yes, 4 � 22.Is the middle term equal to 2(3y)(2)? Yes, 12y � 2(3y)(2).
9y2 � 12y � 4 is a perfect square trinomial.
9y2 � 12y � 4 � (3y)2 � 2(3y)(2) � 22 Write as a2 � 2ab � b2.� (3y � 2)2 Factor using the pattern.
❸❷❶
❸❷❶
Example 1Example 1
Factoring PolynomialsNumber of
Factoring Technique ExampleTerms
2 or more greatest common factor 3x3 � 6x2 � 15x � 3x(x2 � 2x � 5)
2 difference of a2 � b2 � (a � b)(a � b) 4x2 � 25 � (2x � 5)(2x � 5)squares
ax2 � bx � c when m � n � b and mn � ac. � 3x(2x � 1) � 2(2x � 1)
Then use factoring by grouping. � (2x � 1)(3x � 2)
ax � bx � ay � by3xy � 6y � 5x � 10
4 or more factoring by � x(a � b) � y(a � b) � (3xy � 6y) � (5x � 10)
grouping� (a � b)(x � y)
� 3y(x � 2) � 5(x � 2)
� (x � 2)(3y � 5)
Lesson 9-6 Perfect Squares and Factoring 509www.algebra1.com/extra_examples
Lesson 9-6 Perfect Squares and Factoring 509
2 Teach2 Teach
AssessmentDuring the lastlesson of achapter, it isoften good to
review some of the major con-cepts of the chapter to assesswhether students have mas-tered the concepts. The conceptsummary table on FactoringPolynomials provides a perfectopportunity for review. Brieflyreview each factoring techniquewith students. After yourreview, you might considergiving students a quiz on thedifferent techniques to assessstudent mastery.
New
11
In-Class ExampleIn-Class Example
Building on PriorKnowledge
In Chapter 8, students learnedhow to square a binomial such as(a � b)2 or (a � b)2. In this lesson,students will learn how to undothis process to factor perfectsquare trinomials.
FACTOR PERFECT SQUARE TRINOMIALS
Teaching Tip Remind studentsto look closely at the operationsign in front of the second termof the trinomial. This sign signi-fies whether the factors are inthe form (a � b)2 or (a � b)2.
Determine whether eachtrinomial is a perfect squaretrinomial. If so, factor it.
Teaching Tip Point out that4x2 � 36 is indeed a differenceof squares, and can be factoredas (2x � 6)(2x � 6), but remindstudents that polynomials arenot considered to be completelyfactored if the terms have a GCFgreater than 1.
Factor each polynomial.
a. 6x2 � 96 6(x � 4)(x � 4)
b. 16y2 � 8y � 15 (4y � 5)(4y � 3)
SOLVE EQUATIONS WITHPERFECT SQUARES
Teaching Tip Remind studentsthat perfect square trinomialswill always have a repeatedfactor, so they will always haveonly one solution.
Solve 4x2 � 36x � 81 � 0.
�� �9�2
When there is a GCF other than 1, it is usually easier to factor it out first. Then,check the appropriate factoring methods in the order shown in the table. Continuefactoring until you have written the polynomial as the product of a monomialand/or prime polynomial factors.
510 Chapter 9 Factoring
Factor CompletelyFactor each polynomial.
a. 4x2 � 36
First check for a GCF. Then, since the polynomial has two terms, check for thedifference of squares.
4x2 � 36 � 4(x2 � 9) 4 is the GCF.
� 4(x2 � 32) x2 � x • x and 9 � 3 � 3
� 4(x � 3)(x � 3) Factor the difference of squares.
b. 25x2 � 5x � 6
This polynomial has three terms that have a GCF of 1. While the first term is aperfect square, 25x2 � (5x)2, the last term is not. Therefore, this is not a perfectsquare trinomial.
This trinomial is of the form ax2 � bx � c. Are there two numbers m and nwhose product is 25 � �6 or �150 and whose sum is 5? Yes, the product of 15and �10 is �150 and their sum is 5.
� (25x2 � 15x) � (�10x � 6) Group terms with common factors.
� 5x(5x � 3) � 2(5x � 3) Factor out the GCF from each grouping.
� (5x � 3)(5x � 2) 5x � 3 is the common factor.
Solve Equations with Repeated FactorsSolve x2 � x � �
14
� � 0.
x2 � x � �14
� � 0 Original equation
x2 � 2(x)�12
� � �12
�2
� 0 Recognize x2 � x � �14
� as a perfect square trinomial.
x � �12
�2
� 0 Factor the perfect square trinomial.
x � �12
� � 0 Set repeated factor equal to zero.
x � �12
� Solve for x.
Thus, the solution set is ��12
��. Check this solution in the original equation.
Example 3Example 3
Example 2Example 2
SOLVE EQUATIONS WITH PERFECT SQUARES When solving equationsinvolving repeated factors, it is only necessary to set one of the repeated factorsequal to zero.
AlternativeMethodNote that 4x2 � 36 couldfirst be factored as (2x � 6)(2x � 6). Thenthe common factor 2would need to be factoredout of each expression.
Study Tip
510 Chapter 9 Factoring
Logical If students do not understand how a second-degree equationcan have only one solution, suggest that they graph a perfect squaretrinomial on a graphing calculator. The graph will immediately revealhow this is possible. The vertex of the graph of a perfect squaretrinomial equation lies on the x-axis, hence only one solution.
Differentiated Instruction
PowerPoint®
PowerPoint®
Lesson 9-6 Perfect Squares and Factoring 511
Use the Square Root Property to Solve EquationsSolve each equation. Check your solutions.
a. (a � 4)2 � 49
(a � 4)2 � 49 Original equation
a � 4 � ��49� Square Root Property
a � 4 � �7 49 � 7 � 7
a � �4 � 7 Subtract 4 from each side.
a � �4 � 7 or a � �4 � 7 Separate into two equations.
� 3 � �11 Simplify.
The solution set is {�11, 3}. Check each solution in the original equation.
y � 2 � 5 or y � 2 � 5 Separate into two equations.
� 7 � �3 Simplify.
The solution set is {�3, 7}. Check each solution in the original equation.
c. (x � 3)2 � 5
(x � 3)2 � 5 Original equation
x � 3 � ��5� Square Root Property
x � 3 � �5� Add 3 to each side.
Since 5 is not a perfect square, the solution set is �3 � �5��. Using a
calculator, the approximate solutions are 3 � �5� or about 5.24 and
3 � �5� or about 0.76.
Reading Math��36� is read as plus or minus the square rootof 36.
Study Tip
Square Root Property • Symbols For any number n � 0, if x2 � n, then x � ��n�.
• Example x2 � 9
x � ��9� or �3
You have solved equations like x2 � 36 � 0 by using factoring. You can also usethe definition of square root to solve this equation.
x2 � 36 � 0 Original equation
x2 � 36 Add 36 to each side.
x � ��36� Take the square root of each side.
Remember that there are two square roots of 36, namely 6 and �6. Therefore, thesolution set is {�6, 6}. This is sometimes expressed more compactly as {�6}. Thisand other examples suggest the following property.
Example 4Example 4
Lesson 9-6 Perfect Squares and Factoring 511
44
In-Class ExampleIn-Class Example
Teaching Tip Point out tostudents that if they solved x2 � 36 � 0 using factoring,they would also get twosolutions. When the factors (x � 6)(x � 6) are set equal tozero, the solutions are {6, �6}.
Solve each equation. Checkyour solutions.
a. (b � 7)2 � 36 {1, 13}
b. y2 � 12y � 36 � 100 {�16, 4}
c. (x � 9)2 � 8 {�9 2�2�}
PowerPoint®
3 Practice/Apply3 Practice/Apply
Study NotebookStudy NotebookHave students—• complete the definitions/examples
for the remaining terms on theirVocabulary Builder worksheets forChapter 9.
• include explanations on how tofactor perfect square trinomials.
• include any other item(s) that theyfind helpful in mastering the skillsin this lesson.
512 Chapter 9 Factoring
Practice and ApplyPractice and Apply
indicates increased difficulty�
Guided Practice
Application
GUIDED PRACTICE KEYExercises Examples
4, 5 16–11 2
12–16 3, 4
�
�
CHECK You can check your answer using a graphing calculator. Graph y � (x � 3)2
and y � 5. Using the INTERSECT feature ofyour graphing calculator, find where (x � 3)2 � 5. The check of 5.24 as one ofthe approximate solutions is shown at the right.
[�10, 10] scl: 1 by [�10, 10] scl: 1
1. Explain how to determine whether a trinomial is a perfect square trinomial.
2. Determine whether the following statement is sometimes, always, or never true.Explain your reasoning. never; (a � b)2 � a2 � 2ab � b2
a2 � 2ab � b2 � (a � b)2, b 0
3. OPEN ENDED Write a polynomial that requires at least two different factoringtechniques to factor it completely. Sample answer: x3 � 5x2 � 4x � 20
Determine whether each trinomial is a perfect square trinomial. If so, factor it.
16. HISTORY Galileo demonstrated that objects of different weights fall at thesame velocity by dropping two objects of different weights from the top of theLeaning Tower of Pisa. A model for the height h in feet of an object droppedfrom an initial height ho in feet is h � 16t2 � ho, where t is the time in secondsafter the object is dropped. Use this model to determine approximately howlong it took for the objects to hit the ground if Galileo dropped them from aheight of 180 feet. about 3.35 s
Determine whether each trinomial is a perfect square trinomial. If so, factor it.
17–22. See margin.23. GEOMETRY The area of a circle is (16x2 � 80x � 100) square inches. What is
the diameter of the circle? 8x � 20
24. GEOMETRY The area of a square is 81 � 90x � 25x2 square meters. If x is apositive integer, what is the least possible perimeter measure for the square?16 m
Concept Check1. See margin.
Homework HelpFor See
Exercises Examples17–24 125–42 243–59 3, 4
Extra PracticeSee page 841.
512 Chapter 9 Factoring
About the Exercises…Organization by Objective• Factor Perfect Square
Trinomials: 17–22• Solve Equations With
Perfect Squares: 43–48
Odd/Even AssignmentsExercises 17–53 are structuredso that students practice thesame concepts whether theyare assigned odd or evenproblems.
Teaching Tip Remind students thatany of the factoring methods theyhave studied thus far can be used inthe exercises.
Answers1. Determine if the first term is a perfect
square. Then determine if the last termis a perfect square. Finally, check to seeif the middle term is equal to twice theproduct of the square roots of the firstand last terms.
17. no
18. yes; (a � 12)2
19. yes; (2y � 11)2
20. no
21. yes; (3n � 7)2
22. yes; (5a � 12b)2
Study Guide and Intervention
Perfect Squares and Factoring
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
Less
on
9-6
Factor Perfect Square Trinomials
Perfect Square Trinomial a trinomial of the form a2 � 2ab � b2 or a2 � 2ab � b2
The patterns shown below can be used to factor perfect square trinomials.
Squaring a Binomial Factoring a Perfect Square Trinomial
Determine whether 16n2 � 24n � 9 is a perfect squaretrinomial. If so, factor it.Since 16n2 � (4n)(4n), the first term isa perfect square.Since 9 � 3 � 3, the last term is aperfect square.The middle term is equal to 2(4n)(3).Therefore, 16n2 � 24n � 9 is a perfectsquare trinomial.
16n2 � 24n � 9 � (4n)2 � 2(4n)(3) � 32
� (4n � 3)2
Factor 16x2 � 32x � 15.Since 15 is not a perfect square, use a differentfactoring pattern.
16x2 � 32x � 15 Original trinomial
� 16x2 � mx � nx � 15 Write the pattern.
� 16x2 � 12x � 20x � 15 m � �12 and n � �20
� (16x2 � 12x) � (20x � 15) Group terms.
� 4x(4x � 3) � 5(4x � 3) Find the GCF.
� (4x � 5)(4x � 3) Factor by grouping.
Therefore 16x2 � 32x � 15 � (4x � 5)(4x � 3).
Example 1Example 1 Example 2Example 2
ExercisesExercises
Determine whether each trinomial is a perfect square trinomial. If so, factor it.
28. GEOMETRY The area of a circle is given by the formula A � r2, where r is the radius.If increasing the radius of a circle by 1 inch gives the resulting circle an area of 100 square inches, what is the radius of the original circle? 9 in.
29. PICTURE FRAMING Mikaela placed a frame around a print that measures 10 inches by 10 inches. The area of just the frame itself is 69 square inches. What is the width of the frame? 1.5 in.
10
10
x
x
3�5
1�3
9�25
6�5
1�9
2�3
1�5
7�2
1�2
Practice (Average)
Perfect Squares and Factoring
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6Skills Practice, p. 555 and Practice, p. 556 (shown)
Reading to Learn Mathematics
Perfect Squares and Factoring
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
Less
on
9-6
Pre-Activity How can factoring be used to design a pavilion?
Read the introduction to Lesson 9-6 at the top of page 508 in your textbook.
• On the left side of the equation (8 � 2x)2 � 144, the number 8 in the
expression (8 � 2x)2 represents
and 2x represents twice .
• On the right side of the equation, the number 144 represents
in the center of the pavilion, plus
the of the bricks surrounding the center mascot.
Reading the Lesson
1. Three conditions must be met if a trinomial can be factored as a
. Complete the following sentences.
The first term of the trinomial 9x2 � 6x � 1 (is/is not) a perfect square.
The last term of the trinomial, (is/is not) a perfect square.
The is equal to 2(3x)(1).
The trinomial 9x2 � 6x � 1 (is/is not) a trinomial.
2. Match each polynomial from the first column with a factoring technique in the secondcolumn. Some of the techniques may be used more than once. If none of the techniquescan be used to factor the polynomial, write none.
a. 9x2 � 64 iii i. factor as x2 � bx � c
b. 9x2 � 12x � 4 v ii. factor as ax2 � bx � c
c. x2 � 5x � 6 i iii. difference of squares
d. 4x2 � 13x � 9 ii iv. factoring by grouping
e. 9xy � 3y � 6x � 2 iv v. perfect square trinomial
f. x2 � 4x � 4 v vi. factor out the GCF
g. 2x2 � 16 vi
Helping You Remember
3. Sometimes it is easier to remember a set of instructions if you can state them in a shortsentence or phrase. Summarize the conditions that must be met if a trinomial can befactored as a perfect square trinomial. Sample answer: The first and last termsare perfect squares, and the middle term is twice the product of thesquare roots of the first and last terms.
perfect squareismiddle term
isis
perfect square trinomial
areathe area of the school mascot
the width of the pavilionthe length of the pavilion
Reading to Learn Mathematics, p. 557
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
9-69-6
Squaring Numbers: A ShortcutA shortcut helps you to square a positive two-digit number ending in 5.The method is developed using the idea that a two-digit number may be expressed as 10t � u. Suppose u � 5.
(10t � 5)2 � (10t � 5)(10t � 5)
� 100t2 � 50t � 50t � 25
� 100t2 � 100t � 25
(10t � 5)2 � 100t(t � 1) � 25
In words, this formula says that the square of a two-digit number has t(t � 1) in the hundreds place. Then 2 is the tens digit and 5 is the units digit.
FORESTRY For Exercises 55 and 56, use the following information.Lumber companies need to be able to estimate the number of board feet that a givenlog will yield. One of the most commonly used formulas for estimating board feet is
the Doyle Log Rule, B � �1L6�(D2 � 8D � 16), where B is the number of board feet, D is
the diameter in inches, and L is the length of the log in feet.
55. Write this formula in factored form. B � �1L6�(D � 4)2
56. For logs that are 16 feet long, what diameter will yield approximately 256 board feet? 20 in.
FREE-FALL RIDE For Exercises 57 and 58, use the following information.The height h in feet of a car above the exit ramp of an amusement park’s free-fallride can be modeled by h � �16t2 � s, where t is the time in seconds after the cardrops and s is the starting height of the car in feet.
57. How high above the car’s exit ramp should the ride’s designer start the drop in order for riders to experience free fall for at least 3 seconds? 144 ft
58. Approximately how long will riders be in free fall if their starting height is 160 feet above the exit ramp? 3.16 s
s � 3 in.
s in.12s in.
s in.
37. (a2 � 2)(4a � 3b2)
www.algebra1.com/self_check_quiz
�
�
Free-Fall Ride Some amusement parkfree-fall rides can seat4 passengers across per coach and reach speeds of up to 62 miles per hour.Source: www.pgathrills.com
Writing Ask students to look atthe concept summary table onpage 509, and decide whichfactoring technique they likebest. Then have students write adescription of how to use theirfavorite technique, and why theythink it is a better method ascompared to one or two of theother methods.
Assessment Options
Quiz (Lesson 9-6) is available onp. 574 of the Chapter 9 ResourceMasters.
Answer
66. Answers should include thefollowing.
• The length of each side of thepavilion is 8 � x � x or 8 � 2xfeet. Thus, the area of thepavilion is (8 � 2x)2 squarefeet. This area includes the 80 square feet of bricks and the82 or 64-square foot piece of art,for a total area of 144 squarefeet. These two representationsof the area of the pavilion mustbe equal, so we can write theequation (8 � 2x)2 � 144.
514 Chapter 9 Factoring
StandardizedTest Practice
Maintain Your SkillsMaintain Your Skills
Solve each equation. Check your solutions. (Lessons 9-4 and 9-5)
59. HUMAN CANNONBALL Acircus acrobat is shot out of acannon with an initial upwardvelocity of 64 feet per second. If the acrobat leaves the cannon6 feet above the ground, will he reach a height of 70 feet? Ifso, how long will it take him to reach that height? Use themodel for vertical motion. yes; 2 s
CRITICAL THINKING Determine all values of k that make each of the following a perfect square trinomial. 62. 70, �7060. x2 � kx � 64 16, �16 61. 4x2 � kx � 1 4, �4 62. 25x2 � kx � 49
63. x2 � 8x � k 16 64. x2 � 18x � k 81 65. x2 � 20x � k 100
66. Answer the question that was posed at the beginning ofthe lesson. See margin.
How can factoring be used to design a pavilion?
Include the following in your answer:
• an explanation of how the equation (8 � 2x)2 � 144 models the givensituation, and
• an explanation of how to solve this equation, listing any properties used, andan interpretation of its solutions.
67. During an experiment, a ball is dropped off a bridge from a height of 205 feet.The formula 205 � 16t2 can be used to approximate the amount of time, inseconds, it takes for the ball to reach the surface of the water of the river belowthe bridge. Find the time it takes the ball to reach the water to the nearest tenthof a second. C
2.3 s 3.4 s 3.6 s 12.8 s
68. If �a2 � 2�ab � b�2� � a � b, then which of the following statements best describesthe relationship between a and b? D
Vocabulary and Concept CheckVocabulary and Concept Check
www.algebra1.com/vocabulary_review
State whether each sentence is true or false. If false, replace the underlined word ornumber to make a true sentence.
1. The number 27 is an example of a number. false, composite2. is the greatest common factor (GCF) of 12x2 and 14xy. true3. is an example of a perfect square. false, sample answer: 644. 61 is a of 183. true5. The prime factorization for 48 is . false, 24 � 36. x2 � 25 is an example of a . false, difference of squares7. The number 35 is an example of a number. true8. is an example of a prime polynomial. false, sample answer: x2 � 2x � 29. Expressions with four or more unlike terms can be . true
10. is the factorization of a difference of squares. true(b � 7)(b � 7)
factored by grouping
x2 � 3x � 70
composite
perfect square trinomial
3 � 42
factor
66
2xprime
See pages474–479.
9-19-1
ExampleExample
Factors and Greatest Common FactorsConcept Summary
• Prime number: whole number greater than 1 with exactly two factors
• Composite number: whole number greater than 1 with more than two factors
• The greatest common factor (GCF) of two or more monomials is the product of their common factors.
Find the GCF of 15x2y and 45xy2.
15x2y � 3 � 5 � x � x � y Factor each number.
45xy2 � 3 � 3 � 5 � x � y � y Circle the common prime factors.
The GCF is 3 � 5 � x � y or 15xy.
Exercises Find the prime factorization of each integer.See Examples 2 and 3 on page 475.
Have students look through the chapter to make sure they haveincluded questions and notes in their Foldables for each lesson ofChapter 9.Encourage students to refer to their Foldables while completingthe Study Guide and Review and to use them in preparing for theChapter Test.
TM
For more informationabout Foldables, seeTeaching Mathematicswith Foldables.
Lesson-by-LessonReviewLesson-by-LessonReview
Vocabulary and Concept CheckVocabulary and Concept Check
• This alphabetical list ofvocabulary terms in Chapter 9includes a page referencewhere each term wasintroduced.
• Assessment A vocabularytest/review for Chapter 9 isavailable on p. 572 of theChapter 9 Resource Masters.
For each lesson,
• the main ideas aresummarized,
• additional examples reviewconcepts, and
• practice exercises are provided.
The Vocabulary PuzzleMakersoftware improves students’ mathematicsvocabulary using four puzzle formats—crossword, scramble, word search using aword list, and word search using clues.Students can work on a computer screenor from a printed handout.
Vocabulary PuzzleMaker
ELL
MindJogger Videoquizzesprovide an alternative review of conceptspresented in this chapter. Students workin teams in a game show format to gainpoints for correct answers. The questionsare presented in three rounds.
Chapter 9 Study Guide and ReviewChapter 9 Study Guide and Review
ExampleExample
ExampleExample
See pages495–500.
9-49-4
See pages501–506.
9-59-5
Factoring Trinomials: ax2 � bx � c Concept Summary
• Factoring ax2 � bx � c: Find m and n whose product is ac and whose sum is b. Then,write as ax2 � mx � nx � c and use factoring by grouping.
Factor 12x2 � 22x � 14.
First, factor out the GCF, 2: 12x2 � 22x � 14 � 2(6x2 � 11x � 7). In the new trinomial, a � 6, b � 11 and c � �7. Since b is positive, m � n is positive. Since c is negative, mn is negative. So either m or n is negative, but not both. Therefore, make a list of the factors of 6(�7) or �42, where one factor in each pair is negative. Look for a pair of factors whose sum is 11.
� (6x2 � 3x) � (14x � 7) Group terms with common factors.
� 3x(2x � 1) � 7(2x � 1) Factor the GCF from each grouping.
� (2x � 1)(3x � 7) 2x � 1 is the common factor.
Thus, the complete factorization of 12x2 � 22x � 14 is 2(2x � 1)(3x � 7).
42. (2m � 3)(m � 8) 43. (5r � 2)(5r � 2)Exercises Factor each trinomial, if possible. If the trinomial cannot be factoredusing integers, write prime. See Examples 1�3 on pages 496 and 497.
• Difference of Squares: a2 � b2 � (a � b)(a � b) or (a � b)(a � b)
• Sometimes it may be necessary to use more than one factoring techniqueor to apply a factoring technique more than once.
Factor 3x3 � 75x.
3x3 � 75x � 3x(x2 � 25) The GCF of 3x3 and 75x is 3x.
� 3x(x � 5)(x � 5) Factor the difference of squares.
Chapter 9 Study Guide and Review 517
Study Guide and Review
Answers (p. 519)
22. (a � 2)(a � 2)
23. (y � 5)(4m � 3p)
24. 5a(3ab � a � 2)
25. (2y � 3)(3y � 2)
26. 4(s � 5t)(s � 5t)
27. (x � 4)(x � 3)(x � 3)
518 Chapter 9 Factoring
• Extra Practice, see pages 839–841.• Mixed Problem Solving, see page 861.
See pages508–514.
9-69-6
ExamplesExamples
52. ��12
�n � �34
�r���12
�n � �34
�r�
Exercises Factor each polynomial, if possible. If the polynomial cannot befactored, write prime. See Examples 1–4 on page 502.
50. 2y3 � 128y 51. 9b2 � 20 prime 52. �14
�n2 � �196�r2
2y(y � 8)(y � 8)Solve each equation by factoring. Check your solutions. See Example 5 on page 503.
53. b2 � 16 � 0 {�4, 4} 54. 25 � 9y2 � 0 ���53
�, �53
�� 55. 16a2 � 81 � 0 ���94
�, �94
��
Perfect Squares and FactoringConcept Summary
• If a trinomial can be written in the form a2 � 2ab � b2 or a2 � 2ab � b2,then it can be factored as (a � b)2 or as (a � b)2, respectively.
• For a trinomial to be factorable as a perfect square, the first term must be a perfect square, the middle term must be twice the product of the squareroots of the first and last terms, and the last term must be a perfect square.
• Square Root Property: For any number n � 0, if x2 � n, then x � ��n�.
1 Determine whether 9x2 � 24xy � 16y2 is a perfect square trinomial. If so, factor it.
Is the first term a perfect square? Yes, 9x2 � (3x)2.
Is the last term a perfect square? Yes, 16y2 � (4y)2.
Is the middle term equal to 2(3x)(4y)? Yes, 24xy � 2(3x)(4y).
1. Give an example of a prime number and explain why it is prime. Sample 2. Write a polynomial that is the difference of two squares. Then factor your
polynomial. Sample answer: n2 � 100; (n � 10)(n �10)3. Describe the first step in factoring any polynomial. Check for a GCF other than 1 and factor it out.
�, 3�38. GEOMETRY A rectangle is 4 inches wide by 7 inches long. When the length and
width are increased by the same amount, the area is increased by 26 square inches. What are the dimensions of the new rectangle? 6 in. by 9 in.
39. CONSTRUCTION A rectangular lawn is 24 feet wide by 32 feet long. A sidewalk will be built along the inside edges of all four sides. The remaining lawn will have an area of 425 square feet. How wide will the walk be? 3.5 ft
40. STANDARDIZED TEST PRACTICE The area of the shaded part of the square shown at the right is 98 square meters. Find the dimensions of the square. 14 m by 14 m x
x
r
r
r
r 3
3
3
3x
y
www.algebra1.com/chapter_test
Chapter 9 Practice Test 519
Introduction Have you ever noticed that when you are learning the conceptsin a chapter, such as how to factor polynomials in this chapter, that there isoften more than one way to solve a problem?Ask Students Pick a trinomial that can be factored by more than one methodthat you learned in this chapter, and explain how to factor it using thesemethods. Make sure you include a worked-out example with your descriptionsin your portfolio.
Portfolio Suggestion
Assessment Options
Vocabulary Test A vocabularytest/review for Chapter 9 can befound on p. 572 of the Chapter 9Resource Masters.
Chapter Tests There are sixChapter 9 Tests and an Open-Ended Assessment task availablein the Chapter 9 Resource Masters.
Open-Ended AssessmentPerformance tasks for Chapter 9can be found on p. 571 of theChapter 9 Resource Masters. Asample scoring rubric for thesetasks appears on p. A25.
TestCheck andWorksheet Builder
This networkable software hasthree modules for assessment.
• Worksheet Builder to makeworksheets and tests.
• Student Module to take testson-screen.
• Management System to keepstudent records.
Chapter 9 TestsForm Type Level Pages
1 MC basic 559–560
2A MC average 561–562
2B MC average 563–564
2C FR average 565–566
2D FR average 567–568
3 FR advanced 569–570
MC = multiple-choice questionsFR = free-response questions
Standardized Test PracticeStudent Record Sheet (Use with pages 520–521 of the Student Edition.)
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7
2 5 8
3 6 9
Solve the problem and write your answer in the blank.
For Question 12, also enter your answer by writing each number or symbol in abox. Then fill in the corresponding oval for that number or symbol.
10 12
11
12 (grid in)
13
14
15
16
17
Select the best answer from the choices given and fill in the corresponding oval.
18 21
19 22
20
Record your answers for Question 23 on the back of this paper.
DCBA
DCBADCBA
DCBADCBA
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
DCBADCBADCBA
DCBADCBADCBA
DCBADCBADCBA
NAME DATE PERIOD
99
An
swer
s
Part 3 Quantitative ComparisonPart 3 Quantitative Comparison
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 4 Open-EndedPart 4 Open-Ended
Part 1 Multiple ChoicePart 1 Multiple Choice
Standardized Test PracticeStudent Recording Sheet, p. A1
Additional Practice
See pp. 577–578 in the Chapter 9Resource Masters for additionalstandardized test practice.
520 Chapter 9 Factoring
Record your answers on the answer sheetprovided by your teacher or on a sheet ofpaper.
1. Which equation best describes the functiongraphed below? (Lesson 5-3) A
y � ��35
�x � 3
y � �35
�x � 3
y � ��53
�x � 3
y � �53
�x � 3
2. The school band sold tickets to their springconcert every day at lunch for one week. Beforethey sold any tickets, they had $80 in theiraccount. At the end of each day, they recordedthe total number of tickets sold and the totalamount of money in the band’s account.
Which equation describes the relationshipbetween the total number of tickets sold t andthe amount of money in the band’s account a?(Lesson 5-4) D
a � �18
�t � 80 a � �t �
680�
a � 6t � 8 a � 8t � 80
3. Which inequality represents the shadedportion of the graph?(Lesson 6-6) A
y � �13
�x � 1
y �13
�x � 1
y 3x � 1
y � 3x � 1
4. Today, the refreshment stand at the highschool football game sold twice as many bagsof popcorn as were sold last Friday. The totalsold both days was 258 bags. Which system ofequations will determine the number of bagssold today n and the number of bags sold lastFriday f? (Lesson 7-2) D
n � f � 258 n � f � 258f � 2n n � 2f
n � f � 258 n � f � 258f � 2n n � 2f
5. Express 5.387 � 10�3 in standard notation.(Lesson 8-3) B
0.0005387 0.005387
538.7 5387
6. The quotient �186xx4
8�, x 0, is
(Lesson 9-1) C
2x2. 8x2. 2x4. 8x4.
7. What are the solutions of the equation 3x2 � 48 � 0? (Lesson 9-1) A
4, �4 4, �13
�
16, �16 16, �13
�
8. What are the solutions of the equation x2 � 3x � 8 � 6x � 6? (Lesson 9-4) D
2, �7 �2, �4
2, 4 2, 7
9. The area of a rectangle is 12x2 � 21x � 6. The width is 3x � 6. What is the length?(Lesson 9-5) B
4x � 1 4x � 1
9x � 1 12x � 18DC
BA
DC
BA
DC
BA
DCBA
DC
BA
DC
BA
D
C
B
A
y
xO
DC
BA
D
C
B
y
xO
A
Part 1 Multiple Choice
Test-Taking TipQuestions 7 and 9 When answering a multiple-choice question, first find an answer on your own.Then, compare your answer to the answer choicesgiven in the item. If your answer does not matchany of the answer choices, check your calculations.
Monday 12 $176
Tuesday 18 $224
Wednesday 24 $272
Thursday 30 $320
Friday 36 $368
DayTotal Number Total Amount
of Tickets Sold t in Account a
520 Chapter 9 Factoring
These two pages contain practicequestions in the various formatsthat can be found on the mostfrequently given standardizedtests.
A practice answer sheet for thesetwo pages can be found on p. A1of the Chapter 9 Resource Masters.
Log On for Test Practice The Princeton Review offersadditional test-taking tips and
practice problems at their web site. Visitwww.princetonreview.com orwww.review.com
TestCheck andWorksheet Builder
Special banks of standardized testquestions similar to those on the SAT,ACT, TIMSS 8, NAEP 8, and Algebra 1End-of-Course tests can be found onthis CD-ROM.
Open-Ended Assessment ques-tions are graded by using a multi-level rubric that guides you inassessing a student’s knowledgeof a particular concept.
Goal: Write a polynomial anduse it to find the dimensions of adog pen.
Sample Scoring Rubric: The fol-lowing rubric is a sample scoringdevice. You may wish to addmore detail to this sample to meetyour individual scoring needs.
Answers 11. y
xO
x � y � 3
www.algebra1.com/standardized_test
Record your answers on the answer sheetprovided by your teacher or on a sheet ofpaper.
10. Find all values of x that make the equation6x � 2 � 18 true. (Lesson 6-5) 5 and �1
11. Graph the inequality x � y 3. (Lesson 6-6)See margin.
12. A movie theater charges $7.50 for each adultticket and $4 for each child ticket. If thetheater sold a total of 145 tickets for a totalof $790, how many adult tickets were sold?(Lesson 7-2) 60
Compare the quantity in Column A and the quantity in Column B. Then determinewhether:
the quantity in Column A is greater,
the quantity in Column B is greater,
the two quantities are equal, or
the relationship cannot be determinedfrom the information given.
18.
C (Lesson 2-2)
19. B
B (Lesson 3-4)
20.
A (Lesson 5-6)
21.
B (Lesson 7-2)
22.
C (Lesson 9-1)
Record your answers on a sheet of paper.Show your work.
23. Madison is building a fenced, rectangulardog pen. The width of the pen will be3 yards less than the length. The total areaenclosed is 28 square yards. (Lesson 9-4)
a. Using L to represent the length of thepen, write an expression showing thearea of the pen in terms of its length.
b. What is the length of the pen?
c. How many yards of fencing willMadison need to enclose the pencompletely? a–c. See margin.
y
xO
D
C
B
A
Part 2 Short Response/Grid In
Chapter 9 Standardized Test Practice 521
Aligned and verified by
Part 3 Quantitative Comparison
Column A Column B
Column A Column B
the value of x in the value of y in
�23
�x � 27 � 39 �34
�y � 55 � 20
the x-intercept ofthe line that is
the x-intercept ofperpendicular to
the line whose the line graphed
graph is shownabove and passes
through (6, �4)
the y value of the the b value of thesolution of solution of
3x � y � 5 and 2a � 3b � �3 andx � 3y � �6 a � 4b � 24
the GCF of 2x3, the GCF of 18x3,6x2, and 8x 14x2, and 4x
Part 4 Open Ended
x � y if x � �15 x � y if x � �15 and y � �7 and y � �7
Chapter 9 Standardized Test Practice 521
Score Criteria4 A correct solution that is supported
by well-developed, accurateexplanations
3 A generally correct solution, butmay contain minor flaws inreasoning or computation
2 A partially correct interpretationand/or solution to the problem
1 A correct solution with nosupporting evidence or explanation
0 An incorrect solution indicating nomathematical understanding of the concept or task, or no solution is given
23a. a polynomial equationequivalent to 28 � L(L � 3)
23b. The length is 7 yards.
28 � L2 � 3LL2 � 3L � 28 � 0
(L � 7)(L � 4) � 0L � 7
23c. 22 yd
Calculate W. Calculate theperimeter.
W � L � 3W � 7 � 3 or 4P � 2L � 2WP � 2(7) � 2(4)P � 14 � 8 or 22
69. Scientists listening to radio signals would suspect thata modulated signal beginning with prime numberswould indicate a message from an extraterrestrial.
