~ Chapter 9 ~Polynomials and FactoringAlgebra I
Lesson 9-1 Adding & Subtracting Polynomials
Lesson 9-2 Mulitplying and Factoring
Lesson 9-3 Multiplying Binomials
Lesson 9-4 Multiplying Special Cases
Lesson 9-5 Factoring Trinomials of the Type x2 + bx + c
Lesson 9-6 Factoring Trinomials of the Type ax2 + bx + c
Lesson 9-7 Factoring Special Cases
Lesson 9-8 Factoring by Grouping
Chapter Review
Algebra I
Adding & Subtracting Polynomials Cumulative Review
Chap 1-8Lesson 9-1
Adding & Subtracting PolynomialsNotesLesson 9-1
Monomial – an expression that is a number, variable, or a product of a number and one or more variables. (Ex. 8, b, -4mn2, t/3…) (m/n is not a monomial because there is a variable in the denominator)Degree of a Monomial¾ y Degree: 1 ¾ y = ¾ y1… the exponent is 1.3x4y2 Degree: 6 The exponents are 4 and 2. Their sum is 6.-8 Degree: 0 The degree of a nonzero constant is 0.5x0 Degree = ?Polynomial – a monomial or the sum or difference of two or more monomials.Standard form of a Polynomial…Simply means that the degrees of the polynomial terms decrease from left to right.5x4 + 3x2 – 6x + 3 Degree of each?The degree of a polynomial is the same as the degree of the monomial with the greatest exponent. What is the degree of the polynomial above?
Adding & Subtracting PolynomialsNotesLesson 9-1
3x2 + 2x + 1 12 9x4 + 11x 5x5
The number of terms in a polynomial can be used to name the polynomial.
Classifying Polynomials(1)Write the polynomial in standard form.(2) Name the polynomial based on its degree(3) Name the polynomial based on the number of terms6x2 + 7 – 9x4 3y – 4 – y3 8 + 7v – 11vAdding PolynomialsThere are two methods for adding (& subtracting) polynomials…Method 1 – Add vertically by lining up the like terms and adding the
coefficients.Method 2 – Add horizontally by grouping like terms and then adding the
coefficients.(12m2 + 4) + (8m2 + 5) =
Adding & Subtracting PolynomialsNotesLesson 9-1
(9w3 + 8w2) + (7w3 + 4) =Subtracting PolynomialsThere are two methods for subtracting polynomials…Method 1 – Subtract vertically by lining up the like terms and adding
the opposite of each term in the polynomial being subtracted.Method 2 – Subtract horizontally by writing the opposite of each term in
the polynomial being subtracted and then grouping like terms.(12m2 + 4) - (8m2 + 5) =(30d3 – 29d2 – 3d) – (2d3 + d2)
Adding & Subtracting Polynomials
HomeworkLesson 9-1
Homework – Practice 9-1
Multiplying & Factoring Practice 9-1Lesson 9-2
Multiplying & Factoring Practice 9-1Lesson 9-2
Multiplying & Factoring Practice 9-1Lesson 9-2
Mulitplying & FactoringNotesLesson 9-2
Distributing a monomial3x(2x - 3) = 3x(2x) – 3x(3) =-2s(5s - 8) = -2s(5s) – (-2s) (8) =Multiplying a Monomial and a Trinomial
4b(5b2 + b + 6) = 4b(5b2) + 4b(b) + 4b(6) =-7h(3h2 – 8h – 1) = 2x(x2 – 6x + 5) =Factoring a Monomial from a PolynomialFind the GCF for 4x3 + 12x2 – 8x4x3 = 2*2*x*x*x12x2 = 2*2*3*x*x8x = 2*2*2*x What do they all have in common? 2*2*x = 4x
Multiplying & FactoringNotesLesson 9-2
Find the GCF of the terms of 5v5 + 10v3 Find the GCF of the terms of 4b3 – 2b2 – 6bFactoring out a MonomialStep 1: Find the GCFStep 2: Factor out the GCF…Factor 8x2 – 12x =Factor 5d3 + 10d =Factor 6m3 – 12m2 – 24m =Factor 6p6 + 24p5 + 18p3 =
Multiplying & FactoringHomeworkLesson 9-2
Homework ~ Practice 9-2 even
Multiplying Binomials Practice 9-2Lesson 9-3
Multiplying BinomialsNotesLesson 9-3
Using the Distributive PropertySimplify (6h – 7)(2h + 3) = 6h(2h + 3) – 7(2h + 3) =(5m + 2)(8m – 1) = 5m(8m – 1) + 2(8m - 1) =(9a – 8)(7a + 4) = 9a(7a + 4) – 8(7a + 4) =Multiplying using FOIL
F = First O = Outer I = Inner L = Last(6h – 7)(2h + 3) = 6h(2h) + 6h(3) + (-7)(2h) + (-7)(3)
12h2 + 18h + (-14h) + (-21) = 12h2 + 4h -21(3x + 4)(2x + 5) =(3x – 4)(2x – 5) =Applying Multiplication of PolynomialsDetermine the area of each rectangle and subtract the area of center(x + 8)(x + 6) = 3x(x + 3) =
Multiplying BinomialsNotesLesson 9-3
Multiplying a Trinomial and a Binomial
(2x – 3)(4x2 + x -6) = 2x(4x2) + 2x(x) + 2x(-6) -3(4x2) -3(x) -3(-6) 8x3 + 2x2 + (-12x) - 12x2 -3x + 18
Combine like terms = 8x3 – 10x2 – 15x + 18You can also multiply using the vertical multiplication method…Try this one…(6n – 8)(2n2 + n + 7) =
Multiplying BinomialsHomeworkLesson 9-3
Homework – Practice 9-3 even
Multiplying Special CasesPractice 9-3
Lesson 9-4
Multiplying Special CasesPractice 9-3
Lesson 9-4
Multiplying Special CasesNotesLesson 9-4
Finding the Square of a Binomial
(x + 8)2 = (x + 8)(x + 8) =So… (a + b)2 = Rule: The Square of a Binomial (a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Find (t + 6)2
(5y + 1)2
(7m – 2p)2
Find the Area of the shaded region…(x + 4)2 – (x – 1)2
Mental Math – Squares
312 = (30 + 1)2 = 302 + 2(30*1) + 12 = 900 + 60 + 1 = 961
Multiplying Special CasesNotesLesson 9-4
292 =982 =Difference of Squares
(a + b)(a – b) = a2 – ab + ab – b2
= a2 – b2
Find each product.(d + 11)(d – 11) = d2 – 112 = d2 – 121(c2 + 8)(c2 – 8) =(9v3 + w4)(9v3 – w4) =Mental Math
18 * 22 = (20 + 2)(20 – 2) = 202 – 22 = 400 – 4 = 39659 * 61 = 87 * 93 =
Multiplying Special Cases
HomeworkLesson 9-4
Homework – Practice 9-4 odd
Factoring Trinomials of the Type x2 + bx + c
Practice 9-4Lesson 9-5
Factoring Trinomials of the Type x2 + bx + c
Practice 9-4Lesson 9-5
Factoring Trinomials of the Type x2 + bx + cNotesLesson 9-5
Factoring Trinomials
x2 + bx + cTo factor this type of trinomial… you must find two numbers
that have a sum of b and a product of c.Factor x2 + 7x + 12Make a table…Column 1 lists factors of c12… Column 2 lists the sum of those factors… bRow 3 – factors 3 & 4 with a sum of 7 fits so… x2 + 7x + 12 = (x + 3)(x + 4)Factor g2 + 7g + 10Factor a2 + 13a + 30
Factoring Trinomials of the Type x2 + bx + cNotesLesson 9-5
Factoring x2 – bx + cSince the middle term is negative, you must find the negative factors
of c, whose sum is –b.Factor d2 – 17d + 42 > Make a table…Row 3 – factors -3 & -14 with sum of -17So… d2 – 17d + 42 = (d – 3)(d – 14)Factor k2 – 10k + 25Factor q2 – 15q + 36Factoring Trinomials with a negative c (- c)
Factor m2 + 6m - 27Make a tableRow 4 – factors 9 & -3 with sum of 6
Factoring Trinomials of the Type x2 + bx + cNotesLesson 9-5
So… m2 + 6m – 27 = (m + 9)(m – 3)Factor p2 – 3p – 40Factor m2 + 8m – 20Factor y2 – y - 56
Factoring Trinomials of the Type x2 + bx + cHomeworkLesson 9-5
Homework ~ Practice 9-5 #1-30
Factoring Trinomials of the Type ax2 + bx + cPractice 9-5Lesson 9-6
Factoring Trinomials of the Type ax2 + bx + cPractice 9-5Lesson 9-6
Factoring Trinomials of the Type ax2 + bx + cPractice 9-5Lesson 9-6
Factoring Trinomials of the Type ax2 + bx + cNotes
Lesson 9-6
Factoring Trinomials when c is positive
6n2 + 23n + 7… Multiply a & cSo… 6n2 + 2n + 21n + 7 Factor using GCF2n(3n + 1) + 7(3n + 1)(2n + 7)(3n + 1) = 6n2 + 23n + 7Try another one… 2y2 + 9y + 7So… 2y2 + 2y +7y + 7Factor… 2y(y + 1) + 7(y + 1)(2y + 7)(y + 1)What if b is negative? 6n2 – 23n + 76n2 - 2n – 21n + 7
Factors of a*c
Sum (=b)
6 and 7 13
3 and 14 17
2 and 21 23 √
Factoring Trinomials of the Type ax2 + bx + cNotes
Lesson 9-6
2n(3n - 1) – 7(3n – 1)(2n – 7)(3n – 1)Your turn… 2y2 – 5y + 2Factoring Trinomials when c is negative…
7x2 – 26x – 87x2 -28x + 2x – 87x(x – 4) + 2(x – 4)(7x + 2)(x – 4)Factor 5d2 – 14d – 35d2 -15d + 1d – 35d(d – 3) + 1(d – 3)(5d + 1)(d - 3)
Factors of a*c
Sum (=b)
7 and -8 -1
4 and -14 -10
2 and -28 -26 √
Factoring Trinomials of the Type ax2 + bx + cNotes
Lesson 9-6
Factoring Out a Monomial First
20x2 + 80x + 35Factor out the GCF first…5(4x2 + 16x + 7)… then factor 4x2 + 16x + 74x2 + 2x + 14x + 72x(2x + 1) + 7(2x + 1)(2x + 7)(2x + 1) Remember to include the GCF in the final answer5(2x + 7)(2x + 1)Factor 18k2 – 12k - 66(3k2 – 2k – 1)3k2 - 3k + 1k – 13k(k – 1) + 1(k – 1) = 6(3k + 1)(k - 1)
Factoring Trinomials of the Type ax2 + bx + cHomework
Lesson 9-6
Homework: Practice 9-6 first column
Factoring Special CasesPractice 9-6Lesson 9-7
Factoring Special CasesPractice 9-6Lesson 9-7
Factoring Special CasesNotesLesson 9-7
Perfect Square Trinomials
a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2
a2 – 2ab + b2 = (a – b)(a – b) = (a – b)2
So… x2 + 12x + 36 = (x + 6)2
And… x2 – 14x + 49 = (x – 7)2
What about… 4x2 + 12x + 9Factoring a Perfect-Square Trinomial with a = 1 (ax2 + bx + c)x2 + 8x + 16 =n2 – 16n + 64 =Factoring a Perfect-Square Trinomial with a ≠ 1
9g2 – 12g + 44t2 + 36t + 81
Factoring Special CasesNotesLesson 9-7
Factoring the Difference of Squares
a2 – b2 = (a + b)(a – b)Or… x2 – 16 =What about 25x2 – 81 =Try x2 – 36Factor 4w2 – 49Look for common factors…10c2 – 40 =28k2 – 7 =3c4 – 75 =
Factoring Special CasesHomeworkLesson 9-7
Homework: Practice 9-7 odd#1-39
Factoring by GroupingPractice 9-7Lesson 9-8
Factoring by GroupingPractice 9-7Lesson 9-8
Factoring by GroupingPractice 9-7Lesson 9-8
Factoring by GroupingNotesLesson 9-8
Factoring a Four-Term Polynomial
4n3 + 8n2 – 5n – 10Factor the GCF out of each group of 2 terms.? (4n3 + 8n2) - ? (5n + 10)Factor 5t4 + 20t3 + 6t + 24Before you factor, you may need to factor out the GCF.12p4 + 10p3 -36p2 – 30pTry… 45m4 – 9m3 + 30m2 – 6m (factor completely)Finding the dimensions of a rectangular prismThe volume (lwh) of a rectangular prism is 80x3 + 224x2 + 60x. Factor to find the possible expressions for the length, width, and height of the prism.
Factoring by GroupingNotesLesson 9-8
Your turn…Find expressions for possible dimensions of the rectangular prism…V = 6g3 + 20g2 + 16g
V = 3m3 + 10m2 + 3m
Factoring by GroupingHomeworkLesson 9-8
Classwork – Practice 9-8 even # 1-28
Factoring by GroupingPractice 9-8Lesson 9-8
Chap 9 Quiz Review Lesson 7 & 8
Practice 9-8
~ Chapter 9 ~Chapter Review
Algebra I Algebra I
~ Chapter 9 ~Chapter Review
Algebra I Algebra I
