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Name: Date: Instructor: Section:

Copyright © 2016 Pearson Education, Inc. 189

Chapter 5 EXPONENTS AND POLYNOMIALS

5.1 The Product Rule and Power Rules for Exponents

Learning Objectives 1 Use exponents. 2 Use the product rule for exponents.

3 Use the rule ( )m n mna a= .

4 Use the rule ( )m m mab a b= .

5 Use the rule ( )m m

ma ab b

= .

6 Use combinations of the rules for exponents. 7 Use the rules for exponents in a geometry application.

Key Terms

Use the vocabulary terms listed below to complete each statement in exercises 1−3.

exponential expression base power

1. 25 is read “2 to the fifth ______________________”.

2. A number written with an exponent is called a(n) ___________________________.

3. The ________________ is the number being multiplied repeatedly.

Objective 1 Use exponents.

Video Examples

Review these examples for Objective 1: 1. Write 5 5 5⋅ ⋅ in exponential form.

Since 5 occurs as a factor three times, the base is 5 and the exponent is 3.

35 5 5 5⋅ ⋅ =

Now Try: 1. Write 4 4 4 4 4⋅ ⋅ ⋅ ⋅ in

exponential form.

_____________

2. Name the base and exponent of each expression. Then evaluate.

2. Name the base and exponent of each expression. Then evaluate.

a. 43

Base: 3 Exponent: 4

Value: 43 3 3 3 3 81= ⋅ ⋅ ⋅ =

a. 62

_____________


190 Copyright © 2016 Pearson Education, Inc.

b. ( )43-

Base: –3 Exponent: 4

Value: ( ) ( )( )( )( )43 3 3 3 3 81- = - - - - =

b. ( )62-

_____________

Objective 1 Practice Exercises

For extra help, see Examples 1–2 on page 326 of your text.

Write the expression in exponential form and evaluate, if possible.

1. ( )( )( )( )( )1 1 1 1 13 3 3 3 3

1. _________________

Evaluate each exponential expression. Name the base and the exponent.

2. ( )44- 2. ________________

base _____________

exponent _________

3. 83- 3. ________________

base _____________

exponent _________

Objective 2 Use the product rule for exponents.

Video Examples

Review these examples for Objective 2: 3. Use the product rule for exponents to simplify, if

possible.

Now Try: 3. Use the product rule for

exponents to simplify, if possible.

a. 4 58 8⋅

4 5 4 5

9

8 8 8

8

+⋅ =

=

a. 6 79 9⋅

_____________

b. 7 8 9m m m

7 8 9 7 8 9

24

m m m m

m

+ +=

=

b. 11 9 7m m m

_____________



c. 4 9(5 )(6 )x x

( ) ( )4 9 4 9

4 9

13

5 6 5 6

30

30

x x x x

x

x

+

⋅ = ⋅ ⋅ ⋅

=

=

c. 5 6(6 )(3 )x x

_____________

d. 2 35 5+

2 35 5 25 125

150

+ = +=

d. 4 33 3+

_____________


For extra help, see Example 3 on page 327 of your text.

Use the product rule to simplify each expression, if possible. Write each answer in exponential form.

4. 4 37 7⋅ 4. _________________

5. ( )( )7 82 4c c- - 5. _________________

6. ( )( )( )7 2 93 8 2k k k- - 6. _________________

Objective 3 Use the rule ( )m n mna a= .

Video Examples

Review these examples for Objective 3: 4. Use power rule (a) for exponents to simplify.

Now Try: 4. Use power rule (a) for

exponents to simplify.

a. ( )365

( )36 6 3

18

5 5

5

⋅=

=

a. ( )427

_____________

b. ( )43x

( )43 3 4

12

x x

x

⋅=

=

b. ( )65x

_____________





Simplify each expression. Write all answers in exponential form.

7. ( )437 7. ________________

8. ( )94v- 8. ________________

9. ( )73

3é ù-ë û 9. _________________

Objective 4 Use the rule ( )m m mab a b= .

Video Examples

Review this example for Objective 4: 5. Use power rule (b) for exponents to simplify.

Now Try: 5. Use power rule (b) for


( )35xy

( )3 3 3 3

3 3

5 5

125

xy x y

x y

=

=

( )34ab

_____________



Simplify each expression.

10. ( )43 25r t 10. _________________

11. ( )340.2a b- 11. _________________

12. ( )43 72w z- 12. _________________



Objective 5 Use the rule ( )m m

ma ab b

= .

Video Examples

Review this example for Objective 5: 6. Use power rule (c) for exponents to simplify.

Now Try: 6. Use power rule (c) for


( )3

18

( )3 3

31 1 18 5128

= =

( )5

14

_____________



Simplify each expression.

13. ( )3

25x- 13. _________________

14. 4

2xy

z

æ ö÷ç ÷ç ÷çè ø 14. _________________

15. 7

22a

b

æ ö- ÷ç ÷ç ÷çè ø 15. _________________

Objective 6 Use combinations of the rules for exponents.

Video Examples

Review these examples for Objective 6: 7. Simplify each expression.

Now Try: 7. Simplify each expression.

a. ( )3

23 34

⋅

( )3 3 2

23

3 2

3

3 2

3

5

3

3 3 334 14

3 34 1

34

3 243, or 644

+

⋅ = ⋅

⋅=⋅

=

=

a. ( )3

25 52

⋅

_____________



b. ( ) ( )4 35 6 5x y x y- -

( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( )( )

4 35 6 5

4 35 6 5

4 3 34 35 4 6 5

4 320 4 18 15

7 20 18 4 15

38 19

38 19

1 1

1 1

1 1

1

1

x y x y

x y x y

x y x y

x y x y

x y

x y

x y

+ +

- -

= - -

= - ⋅ -

= - ⋅ -

= -

=-

=-

b. ( ) ( )3 25 6 5x y x y- -

_____________


For extra help, see Example 7 on page 330–331 of your text.

Simplify. Write all answers in exponential form.

16. ( ) ( )2 43 5x x- - 16. _________________

17. ( ) ( )5 422ab c ab 17. _________________

18. ( ) ( )7 42 3 45 5x y xy 18. _________________



Objective 7 Use the rules for exponents in a geometry application.

Video Examples

Review this example for Objective 7: 8. Find the area of the figure.

Now Try: 8. Find the area of the figure.

45m

625

m

Use the formula for the area of a rectangle.

( )( )4 6

4 6

10

255

255

2

A LW

A m m

A m

A m

+

=

=

= ⋅ ⋅

=

37x

24x

_____________



Find a polynomial that represents the area of each figure.

19.

19. _________________

20.

20. _________________

21. 21. _________________




5.2 Integer Exponents and the Quotient Rule

Learning Objectives 1 Use 0 as an exponent. 2 Use negative numbers as exponents. 3 Use the quotient rule for exponents. 4 Use combinations of the rules for exponents.

Key Terms


exponent base product rule for exponents

power rule for exponents

1. The statement “If m and n are any integers, then ( )nm mna a= ” is an example of the _______________________________.

2. In the expression ma , a is the _______________ and m is the _______________.

3. The statement “If m and n are any integers, then m n m na a a +⋅ = is an example of the _______________________________.

Objective 1 Use 0 as an exponent.

Video Examples

Review these examples for Objective 1: 1. Evaluate.

Now Try: 1. Evaluate.

a. 075 1=

a. 088

_____________

b. ( )075 1 or 1- =- -

b. 088-

_____________

c. ( ) ( )09 1 0x x- = ¹

c. ( ) ( )088 0a a- ¹

_____________

d. 0 03 12 1 1 2+ = + =

d. 0 05 16-

_____________





Evaluate each expression.

1. 012- 1. _________________

2. ( )0015 15- - - 2. _________________

3. 8

008

3. _________________

Objective 2 Use negative numbers as exponents.

Video Examples

Review these examples for Objective 2: 2. Simplify by writing with positive exponents.

Assume that all variables represent nonzero real numbers.

Now Try: 2. Simplify by writing with

positive exponents. Assume that all variables represent nonzero real numbers.

a. 34-

33

1 14 , or 644

- =

a. 33-

_____________

b. ( )3

13

-

( )3

31 3 , or 273

-=

b. ( )2

15

-

_____________

c. ( )5

23

-

( ) ( )5 5

5

5

2 33 2

3224332

-=

=

=

c. ( )33

2

-

_____________



d. 1 15 3- --

1 1 1 15 35 33 5

15 152

15

- -- = -

= -

=-

d. 1 14 8- --

_____________

e. 3 ( 0)q q- ¹

33

1qq

- =

e. 5 ( 0)p p- ¹

_____________

3. Simplify. Assume that all variables represent nonzero real numbers.

3. Simplify. Assume that all variables represent nonzero real numbers.

a. 4 2

2 43 7 49, or

817 3

-

- = a. 2

365

-

-

_____________

b. 4

6 46

ba ba

- =

b. 7 2x y-

_____________

c. 3 4

4 34 4

x y yz

z x

-

- = c. 3

54

p q

r

-

-

_____________


For extra help, see Examples 2–3 on pages 336–337 of your text.

Evaluate or simplify each expression, and write it using only positive exponents. Assume that all variables represent nonzero real numbers.

4. 42k-- 4. _________________

5. 2 9( )m n - 5. _________________

6. 4

723

xy

-

- 6. _________________



Objective 3 Use the quotient rule for exponents.

Video Examples

Review these examples for Objective 3: 4. Simplify. Assume that all variables represent

nonzero real numbers.

Now Try: 4. Simplify. Assume that all

variables represent nonzero real numbers.

a. 9

9 6 36

4 4 4 644

-= = = a. 18

1633

_____________

b. ( )

66 4 10

4p

p pp

- -- = = b.

6

4z

z-

_____________

c. ( )( )

3

57

7

x

x

-

-++

7x ¹-

( )( )

( ) ( )

( )

( )

33 5

5

3 5

2

7 77

7

7

x xx

x

x

-- - -

-

- +

+ = ++

= +

= +

c. ( )

( )

8

10a b

a b

-

---

a b¹

_____________

d. 4 3

1 3 48

5

x y

x y

-

- -

4 3 3 4

1 3 4 3 4

7

7

8 8 5

5

40

x y y y

x y x x

y

x

-

- -⋅=

=

d. 5 3

1 3 49

4a ba b

-

- -

_____________


For extra help, see Example 4 on pages 338–339 of your text.

Use the quotient rule to simplify each expression, and write it using only positive exponents. Assume that all variables represent nonzero real numbers.

7. 7 10

3 548k mk m

7. _________________



8. 4 3

2 3a b

a b- - 8. _________________

9. 1 4 6

4 1 23

3

m p

m p

- -

- - 9. _________________

Objective 4 Use combinations of the rules for exponents.

Video Examples

Review these examples for Objective 4: 5. Simplify each expression. Assume that all

variables represent nonzero real numbers.

Now Try: 5. Simplify each expression.

Assume that all variables represent nonzero real numbers.

a. ( )24

655

( )24 8

6 6

8 6

2

5 55 5

5

5

25

-

=

=

==

a. ( )23

566

_____________

b. ( ) ( )4 23 3a a

( ) ( ) ( )4 2 6

6 6

6

3 3 3

3

729

a a a

a

a

=

=

=

b. ( ) ( )3 25 5b b

_____________

c. 543

4x

-æ ö÷ç ÷ç ÷çè ø

5 54

4

5

5 20

20

3 44 3

431024

243

xx

x

x

-æ ö æ ö÷ç ÷=ç÷ç ÷ç ÷ç ÷è ø è ø

=

=

c. 542

3p

-æ ö÷ç ÷ç ÷çè ø

_____________



d. ( )( )

52 3

62 43

k m n

km n

--

--

( )( )

5 2 5 3 5 52 3

6 6 6 2 6 4 62 4

10 15 5

6 6 12 24

6 15 12

6 10 24 5

27

4 29

( ) ( )

3 ( ) ( )3

3

3

729

k m nk m nk m nkm n

k m nk m n

mk n

mk n

- - - - --

- - - - - --

- -

- - -

+

- + +

=

=

=

=

d. ( )( )

32 3

44 2

7xy z

x yz

--

- -

_____________



Simplify each expression, and write it using only positive exponents. Assume that all variables represent nonzero real numbers.

10. ( ) ( )7 89 9xy xy

- 10. _________________

11. ( ) ( )

( )

4 61 2 2

23

a b ab

a b

-- -

- 11. _________________

12. 43 4

2 1k t

k t

-

-

æ ö÷ç ÷ç ÷÷çè ø 12. _________________




5.3 Scientific Notation

Learning Objectives 1 Express numbers in scientific notation. 2 Convert numbers in scientific notation standard notation. 3 Use scientific notation in calculations.

Key Terms


scientific notation quotient rule power rule

1. A number written as 10na´ , where 1 10a£ < and n is an integer, is written in

____________________________.

2. The statement “If m and n are any integers and b ≠ 0, then ( )m m

ma ab b

= ” is an

example of the _______________________________.

3. The statement “If m and n are any integers and b ≠ 0, then m

m nn

a aa

-= ” is an

example of the _______________________________.

Objective 1 Express numbers in scientific notation.

Video Examples

Review these examples for Objective 1: 1. Write each number in scientific notation.

Now Try: 1. Write each number in scientific

notation. a. 84,300,000,000

Move the decimal point 10 places to the left.

1084,300,000,000 8.43 10= ´

a. 47,710,000,000

_____________

b. 0.00573

The first nonzero digit is 5. Count the places. Move the decimal point 3 places to the right.

30.00573 5.73 10-= ´

b. 0.0463

_____________





Write each number in scientific notation.

1. 23,651 1. _________________

2. −429,600,000,000 2. _________________

3. −0.0002208 3. _________________

Objective 2 Convert numbers in scientific notation standard notation.

Video Examples

Review these examples for Objective 2: 2. Write each number without exponents.

Now Try: 2. Write each number without

exponents.

a. 63.57 10´

Move the decimal point 6 places to the right, and attach four zeros.

63.57 10 3,570,000´ =

a. 72.796 10´

_____________

b. 38.98 10-´

Move the decimal point 3 places to the left.

38.98 10 0.00898-´ =

b. 41.64 10-´

_____________



Write each number in standard notation.

4. 62.45 10- ´ 4. ________________

5. 36.4 10-´ 5. _________________

6. 44.02 10- ´ 6. _________________



Objective 3 Use scientific notation in calculations.

Video Examples

Review these examples for Objective 3: 3. Perform each calculation. Write answers in

scientific notation and also without exponents.

Now Try: 3. Perform each calculation. Write

answers in scientific notation and also without exponents.

a. ( )( )4 38 10 7 10´ ´

( )( ) ( )( )

( )

4 3 4 3

7

1 7

8

8 10 7 10 8 7 10 10

56 10

5.6 10 10

5.6 10

560,000,000

´ ´ = ´ ´

= ´

= ´ ´

= ´=

a. ( )( )5 29 10 3 10´ ´

_____________ _____________

b. 4

26 103 10

-´´

4 4

2 2

6

6 10 6 1033 10 10

2 10

0.000002

- -

-

´ = ´´

= ´=

b. 3

539 1013 10

-´´

_____________ _____________

4. The Sahara desert covers approximately 63.5 10´ square miles. Its sand is, on average,

12 feet deep. Find the volume, in cubic feet, of

sand in the Sahara. ( )2 2 2Hint: 1 mi 5280 ft= Round your answer to two decimal places.

( )( )( )

( )

6 2

6

6

15 3

3.5 10 5280 12

97574400 12 10

1170892800 10

1.17 10 ft

´

= ´

= ´

» ´

The volume is 151.17 10´ cubic feet.

4. The Sahara desert covers

approximately 63.5 10´ square miles. Its sand is, on average, 12 feet deep. The volume of a single grain of sand is

approximately 91.3 10-´ cubic feet. About how many grains of sand are in the Sahara?

_____________





Perform the indicated operations, and write the answers in scientific notation.

7. ( ) ( )4 22.3 10 1.1 10-´ ´ ´ 7. _________________

8. 1

39.39 10

3 10´

´ 8. _________________

Work the problem. Give answer in scientific notation.

9. There are about 236 10´ atoms in a mole of atoms.

How many atoms are there in 58.1 10-´ mole?

9. _________________




5.4 Adding, Subtracting, and Graphing Polynomials

Learning Objectives 1 Identify terms and coefficients. 2 Combine like terms. 3 Know the vocabulary for polynomials. 4 Evaluate polynomials. 5 Add and subtract polynomials. 6 Graph equations defined by polynomials of degree 2.

Key Terms


term like terms polynomial

descending powers degree of a term

degree of a polynomial monomial

binomial trinomial

parabola vertex axis line of symmetry

1. The _________________________ is the sum of the exponents on the variables in that term.

2. A polynomial in x is written in ___________________________ if the exponents on x in its terms are decreasing order.

3. A ________________ is a number, a variable, or a product or quotient of a number and one or more variables raised to powers.

4. A polynomial with exactly three terms is called a ________________________.

5. A ____________________ is a term, or the sum of a finite number of terms with whole number exponents.

6. A polynomial with exactly one term is called a ______________________.

7. The ____________________________ is the greatest degree of any term of the polynomial.

8. A ____________________ is a polynomial with exactly two terms.

9. Terms with exactly the same variables (including the same exponents) are called ___________________________.



10. If a graph is folded on its_______________________________, the two sides coincide.

11. The _______________________________ of a parabola that opens upward or downward is the lowest or highest point on the graph.

12. The _______________________________ of a parabola that opens upward or downward is a vertical line through the vertex.

13. The graph of the quadratic equation 2y ax bx c= + + is called a __________________.

Objective 1 Identify terms and coefficients.

Video Examples

Review this example for Objective 1: 1. For each expression, determine the number of

terms and name the coefficients of the terms.

Now Try: 1. For each expression, determine

the number of terms and name the coefficients of the terms.

4 26 3x x- -

Rewrite the expression as 0 4 26 3 1x x x- - .

There are three terms: 4 26, 3 , and .x x- - The coefficients are 6, –3, and –1.

2 7 2x x+ -

_____________



For each expression, determine the number of terms and name the coefficients of the terms.

1. 23 2x x- + 1. _________________

2. 35 6z+ 2. _________________

3. 38 1y y- - 3. _________________



Objective 2 Combine like terms.

Video Examples

Review this example for Objective 2: 2. Simplify the expression by combining like terms.

Now Try: 2. Simplify the expression by

combining like terms.

3 319 6 5m m m+ +

( )3 3 3

3

19 6 5 19 5 6

24 6

m m m m m

m m

+ + = + +

= +

2 3 222 15 7m m m+ +

_____________



In each polynomial, combine like terms whenever possible. Write the result with descending powers.

4. 3 3 3 37 4 5 11z z z z- + - 4. _________________

5. 7 7 81.3 0.4 2.6z z z- + + 5. _________________

6. 3 2 2 2 36 9 2 14 3 6 8 2c c c c c c- - + + - - + 6. _________________

Objective 3 Know the vocabulary for polynomials.

Video Examples

Review these examples for Objective 3: 3. Simplify each polynomial, if possible, and write

in descending powers of the variable. Then give the degree and tell whether the polynomial is a monomial, a binomial, a trinomial, or none of these.

Now Try: 3. Simplify each polynomial, if

possible, and write in descending powers of the variable. Then give the degree and tell whether the polynomial is a monomial, a binomial, a trinomial, or none of these.

a. 3 25 6 7 2 4x x x x x+ - - +

3 2 3 25 6 7 2 4 6 2 2x x x x x x x x+ - - + = - + The degree is 3. The simplified polynomial is a trinomial.

a. 2 32 7 6 3 8x x x x x- - + +

_____________



b. 3 5 3 59 7 3 2y y y y- + +

3 5 3 5 5 39 7 3 2 5 12y y y y y y- + + =- + The degree is 5. The simplified polynomial is a binomial.

b. 5 2 5 24 3w w w w- + +

_____________



For each polynomial, first simplify, if possible, and write the resulting polynomial in descending powers of the variable. Then give the degree of this polynomial, and tell whether it is a monomial, a binomial, a trinomial, or none of these.

7. 8 2 83 2n n n- - 7. ________________

degree: __________

type: ____________

8. 2 3 8 53.2 5.7 1.1d d d d- + - - 8. ________________

degree: __________

type: ____________

9. 4 2 4 2 56 6 9 4 5c c c c c- - + - + 9. ________________

degree: __________

type: ____________

Objective 4 Evaluate polynomials.

Video Examples

Review this example for Objective 4:

4. Find the value of 3 24 6 5 5x x x+ - - for

Now Try: 4. Find the value of

4 25 3 9 7x x x+ - - for 2x =

( ) ( ) ( )

( ) ( ) ( )

3 2

3 2

4 6 5 5

4 2 6 2 5 2 5

4 8 6 4 5 2 5

32 24 10 5

41

x x x+ - -

= + - -

= + - -= + - -=

4x =

_____________





Find the value of each polynomial (a) when x = –2 and (b) when x = 3.

10. 33 4 19x x+ - 10. a. _______________

b. _______________

11. 3 24 10 1x x- + - 11. a. _______________

b. _______________

12. 4 23 8 9x x x- - + 12. a. _______________

b. _______________

Objective 5 Add and subtract polynomials.

Video Examples

Review these examples for Objective 5: 6. Find each sum.

Now Try: 6. Find each sum.

a. Add 4 35 7 9x x- + and 4 33 8 7x x- + -

( ) ( )4 3 4 3

4 4 3 3

4 3

5 7 9 3 8 7

5 3 7 8 9 7

2 2

x x x x

x x x x

x x

- + + - + -

= - - + + -

= + +

a. Add 315 5 3x x- + and 311 6 9x x- + +

_____________

b. ( ) ( )4 2 3 25 7 6 3 4 7x x x x x- + + - + -

( ) ( )4 2 3 2

4 3 2 2

4 3 2

5 7 6 3 4 7

5 3 7 4 6 7

5 3 3 6 7

x x x x x

x x x x x

x x x x

- + + - + -

= - - + + -

= - - + -

b. ( ) ( )2 38 6 4 7 8 5x x x x- + + - -

_____________



7. Perform the subtraction. 7. Perform the subtraction.

Subtract 3 28 5 8x x- + from 3 29 6 7x x+ - .

( ) ( )( ) ( )

3 2 3 2

3 2 3 2

3 2

9 6 7 8 5 8

9 6 7 8 5 8

11 15

x x x x

x x x x

x x

+ - - - +

= + - + - + -

= + -

Subtract 318 4 6x x+ - from 37 3 5x x- - .

_____________

9. Perform the indicated operations to simplify the expression

( ) ( ) ( )2 2 25 2 9 7 5 8 6 3 5x x x x x x- + - - + + + -

Rewrite, changing the subtraction to adding the opposite. ( ) ( ) ( )( ) ( ) ( )( ) ( )

2 2 2

2 2 2

2 2

2

5 2 9 7 5 8 6 3 5

5 2 9 7 5 8 6 3 5

2 3 6 3 5

4 6 4

x x x x x x

x x x x x x

x x x x

x x

- + - - + + + -

= - + + - + - + + -

= - + + + + -

= + -

9. Perform the indicated operations to simplify the expression ( ) ( )( )

2 2

2

10 7 6 5 11 3

2 4 7

x x x x

x x

- + - - +

+ + -

_____________

10. Add or subtract as indicated. 10. Add or subtract as indicated.

( ) ( )2 2 2 23 5 4 3x y xy y x y xy y+ + - + -

Change the signs of the terms in the parentheses and add like terms vertically.

2 2

2 2

2 2

3 5

4 3

4 4

x y xy y

x y xy y

x y xy y

+ +

- - +

- + +

( )( )

2 2

2 2

7 3 4

6 4

x y xy y

x y xy y

+ +

- - +

_____________



Add or subtract as indicated.

13. ( ) ( )3 2 23 5 6 2 5 4r r r r+ - + - + 13. _________________



14. ( ) ( )3 2 28 11 12 10 3w w w- + - - - + 14. _________________

15. ( ) ( ) ( )2 2 2 22 2 4 6 9 9 5x y xy xy xy xy x y xy+ - + + - + 15. _________________

Objective 6 Graph equations defined by polynomials of degree 2.

Video Examples

Review this example for Objective 6: 11. Graph the equation.

Now Try: 11. Graph the equation.

2 3y x= -

Find several ordered pairs. Let x = 0 to find the y-intercept.

2 23 0 3 3y x= - = - =- This gives the ordered pair (0, –3). Select several values for x and find the corresponding values for y. Plot the ordered pairs and join them with a smooth curve.

2 1

1 2

0 3

1 2

2 1

x y

--

- --

29y x= -





Graph each equation.

16. 2 1y x 16.

vertex: _______________________

17. 2 2y x 17.

vertex: _______________________




5.5 Multiplying Polynomials

Learning Objectives 1 Multiply a monomial and a polynomial. 2 Multiply two polynomials. 3 Multiply binomials by the FOIL method.

Key Terms


FOIL outer product inner product

1. The ____________________________ of ( )( )2 5 8y y- + is 5y- .

2. ___________________ is a shortcut method for finding the product of two binomials.

3. The ____________________________ of ( )( )2 5 8y y- + is 16y .

Objective 1 Multiply a monomial and a polynomial.

Video Examples

Review this example for Objective 1: 1. Find the product.

Now Try: 1. Find the product.

( )25 7 3x x+

Use the distributive property.

( ) ( ) ( )2 2 2

3 2

5 7 3 5 7 5 3

35 15

x x x x x

x x

+ = +

= +

( )38 4 8x x+

_____________



Find each product.

1. ( )37 5 2z z + 1. _________________

2. ( )2 32 3 7 3m m m+ + 2. _________________

3. ( )2 3 23 2 3 4 11y y y y- + - + 3. ________________



Objective 2 Multiply two polynomials.

Video Examples

Review these examples for Objective 2:

2. Multiply ( )( )2 3 26 5 4 3x x x x+ - + .

Multiply each term of the second polynomial by each term of the first.

( )( )( ) ( ) ( )

( ) ( ) ( )

2 3 2

2 3 2 2 2

3 2

5 4 3 3 2

5 4 3 2

6 5 4 3

5 4 3

6 5 6 4 6 3

5 4 3 30 24 18

5 4 33 24 18

x x x x

x x x x x x

x x x

x x x x x x

x x x x x

+ - +

= + - +

+ + - +

= - + + - +

= - + - +

Now Try: 2. Multiply

( )( )3 4 29 4 2x x x x+ - +

_____________

3. Multiply ( )( )3 22 7 5 1 4 6x x x x+ + - + vertically.

Write the polynomials vertically.

3 22 7 5 1

4 6x x x

x+ + -

+

Begin by multiplying each term in the top row by 6.

3 2

3 2

2 7 5 14 6

12 42 30 6

x x xx

x x x

+ + -+

+ + -

Now multiply each term in the top row by 4x. Then add like terms.

3 2

3 2

4 3 2

4 3 2

2 7 5 14 6

12 42 30 6

8 28 20 4

8 40 62 26 6

x x xx

x x x

x x x x

x x x x

+ + -+

+ + -+ + -+ + + -

The product is 4 3 28 40 62 26 6x x x x+ + + - .

3. Multiply

( )( )3 24 3 6 5 7 3x x x x- + + - vertically.

_____________



4. Find the product of 3 216 12 4m m- + + and 21 3

4 4m + .

Multiply each term of the second polynomial by each term of the first.

( )( )( ) ( ) ( )( ) ( ) ( )

3 2 2

3 2 3 2 2

2 2

3 2 5 4 2

5 4 3 2

1 316 12 44 4

1 3 116 16 124 4 4

3 1 3 12 4 44 4 4

12 9 3 5 3

4 3 12 10 3

m m m

m m m m m

m m

m m m m m

m m m m

- + + +

=- - +

+ + +

=- + + - + +

=- + - + +

The product is 5 4 3 24 3 12 10 3m m m m- + - + + .

4. Find the product of 3 212 36 6x x- + and 21 5

6 6x + .

_____________



Find each product.

4. ( )( )23 3 9x x x+ - + 4. ________________

5. ( )( )2 3 22 1 3 2 4m m m m+ + - 5. _________________

6. ( )( )2 23 2 3 4x x x x+ + - 6. _________________

Objective 3 Multiply binomials by the FOIL method.

Video Examples

Review these examples for Objective 3: 5. Use the FOIL method to find the product

( )( )7 5x x+ - .

Step 1 F Multiply the first terms: ( ) 2x x x= .

Now Try: 5. Use the FOIL method to find the

product ( )( )9 6x x+ - .

_____________



Step 2 O Find the outer product: ( )5 5x x- =- .

Step 3 I Find the inner product: ( )7 7x x= . Add the outer and inner products mentally: 5 7 2x x x- + =

Step 4 L Multiply the last terms: ( )7 5 35- =- .

The product ( )( )7 5x x+ - is 2 2 35x x+ - .

6. Multiply ( )( )7 3 4 5x y- + .

First ( )7 4 28x y xy=

Outer ( )7 5 35x x=

Inner ( )3 4 12y y- =-

Last ( )3 5 15- =-

The product ( )( )7 3 4 5x y- + is

28 35 12 15xy x y+ - - .

6. Multiply ( )( )8 7 2 9y x- + .

_____________

7. Find the product. 7. Find the product.

( )( )3 7 2 9k m k m+ +

( )( )

( ) ( ) ( ) ( )

2 2

2 2

3 7 2 9

3 2 3 9 7 2 7 9

6 27 14 63

6 41 63

k m k m

k k k m m k m m

k km km m

k km m

+ +

= + + +

= + + +

= + +

( )( )5 8 3 4k n k n+ +

_____________



Find each product.

7. ( )( )5 4 3a b a b- + 7. _________________

8. ( )( )3 4 1 2a a+ + 8. _________________

9. ( )( )2 3 3 4m n m n+ - + 9. _________________




5.6 Special Products

Learning Objectives 1 Square binomials. 2 Find the product of the sum and difference of two terms. 3 Find greater powers of binomials.

Key Terms


conjugate binomial

1. A polynomial with two terms is called a __________________________.

2. The ___________________ of a + b is a − b.

Objective 1 Square binomials.

Video Examples

Review these examples for Objective 1: 2. Square each binomial.

Now Try: 2. Square each binomial.

a. ( )26 3x y+

( ) ( ) ( )( ) ( )2 22

2 2

6 3 6 2 6 3 3

36 36 9

x y x x y y

x xy y

+ = + +

= + +

a. ( )22 9a k+

_____________

b. ( )216

4n+

( ) ( ) ( )( ) ( )

2 22

2

1 1 16 6 2 64 4 4

136 316

n n n

n n

+ = + +

= + +

b. ( )2

136

p+

_____________



Find each square by using the pattern for the square of a binomial.

1. ( )27 x+ 1. _________________



2. ( )22 3m p- 2. _________________

3. ( )24 0.7y- 3. ________________

Objective 2 Find the product of the sum and difference of two terms.

Video Examples

Review these examples for Objective 2: 3. Find each product.

Now Try: 3. Find each product.

a. ( )( )5 5x x+ -

Use the rule for the product of the sum and difference of two terms.

( )( ) 2 2

2

5 5 5

25

x x x

x

+ - = -

= -

a. ( )( )9 9x x+ -

_____________

b. ( )( )3 34 4

y y- +

( )( ) ( )

22

2

3 3 34 4 4

916

y y y

y

- + = -

= -

b. ( )( )5 56 6

a a+ -

_____________

4. Find each product. 4. Find each product.

a. ( )( )6 6x w x w+ -

( )( ) ( )2 2

2 2

6 6 6

36

x w x w x w

x w

+ - = -

= -

a. ( )( )11 11x y x y- +

_____________

b. ( )( )2 23 4 4q q q+ -

First, multiply the conjugates.

( )( ) ( )2 2 4

5

3 4 4 3 16

3 48

q q q q q

q q

+ - = -

= -

b. ( )( )2 24 6 6p p p+ -

_____________





Find each product by using the pattern for the sum and difference of two terms.

4. ( )( )12 12x x+ - 4. ________________

5. ( )( )8 5 8 5k p k p+ - 5. ________________

6. ( )( )4 47 7

2 2t u t u+ - 6. ________________

Objective 3 Find greater powers of binomials.

Video Examples

Review these examples for Objective 3: 5. Find each product.

Now Try: 5. Find each product.

a. ( )34x+

( )

( ) ( )

( )( )

3

2

2

3 2 2

3 2

4

4 4

8 16 4

8 16 4 32 64

12 48 64

x

x x

x x x

x x x x x

x x x

+

= + +

= + + +

= + + + + +

= + + +

a. ( )36x+

_____________

b. ( )45 4y-

( )

( ) ( )

( )( )

4

2 2

2 2

4 3 2

3 2

2

4 3 2

5 4

5 4 5 4

25 40 16 25 40 16

625 1000 400

1000 1600 640

400 640 256

625 2000 2400 1280 256

y

y y

y y y y

y y y

y y y

y y

y y y y

-

= - -

= - + - +

= - +

- + -

+ - +

= - + - +

b. ( )43 5x-

_____________





Find each product.

7. ( )33a- 7. _________________

8. ( )43j+ 8. _________________

9. ( )44 3s t+ 9. _________________




5.7 Dividing Polynomials

Learning Objectives 1 Divide a polynomial by a monomial. 2 Divide a polynomial by a polynomial. 3 Use division in a geometry problem.

Key Terms


quotient dividend divisor

1. In the division 5 3

32

5 10 25

x x x xx- = - , the expression 5 35 10x x- is the

__________________.


32

5 10 25

x x x xx- = - , the expression 3 2x x- is the __________.


32

5 10 25

x x x xx- = - , the expression 25x is the _____________.

Objective 1 Divide a polynomial by a monomial.

Video Examples


1. Divide 4 36 18x x- by 26x .

4 3 4 3

2 2 2

2

6 18 6 186 6 6

3

x x x xx x x

x x

- = -

= -

Check Multiply. ( )2 2 4 36 3 6 18x x x x x- = -

Now Try:

1. Divide 4 220 10x x- by 2x .

_____________

2. Divide. 6 4 2

325 15 10

5a a a

a- +

Divide each term by 35a . 6 4 2 6 4 2

3 3 3 3

3

25 15 10 25 15 105 5 5 5

25 3

a a a a a aa a a a

a aa

- + = - +

= - +

2. Divide. 5 4 2

327 36 18

9n n n

n- -

_____________



3. Divide 4 512 15 5x x x- + - by 5 .x-

Write the polynomial in descending powers before dividing.

5 4 5 4

4 3

15 12 5 15 12 55 5 5 5

123 15

x x x x x xx x x x

x x

- - = - -- - - -

=- + +

Check ( )( ) ( ) ( )

4 3

4 3

5 4

125 3 15

125 3 5 5 15

15 12 5

x x x

x x x x x

x x x

- - + +

=- - - -

= - -

3. Divide 5 68 7 10 6z z z- + - - by 22 .z

_____________

4. Divide 5 9 3 7 2 5 3 2225 150 110 80 75x y x y x y xy y- + - +

by 225 .xy-

5 9 3 7 2 5 3 2

2

225 150 110 80 75

25

x y x y x y xy y

xy

- + - +-

5 9 3 7 2 5 3 2

2 2 2 2 2

225 150 110 80 75

25 25 25 25 25

x y x y x y xy y

xy xy xy xy xy= - + - +

- - - - -3

4 7 2 5 22 16 39 65 5xy y

x y x yx

=- + - + -

Check by multiplying the quotient by the divisor.

4. Divide 5 3 4 2 280 160 120a b a b a b+ - by

240a b- .

_____________



Perform each division.

1. 5 3

216 24

8a a

a- 1. _________________

2. 5 4 3

312 28 8 3

4z z z z

z+ - + 2. _________________

3. 4 3

239 12 15

3m m

m- +-

3. _________________



Objective 2 Divide a polynomial by a polynomial.

Video Examples


5. Divide 22 11 15

3x x

x- +-

.

Step 1 22x divided by x is 2x ( ) 22 3 2 6x x x x- = -

Step 2 Subtract. Bring down the next term.

Step 3 5x- divided by x is –5. ( )5 3 5 15x x- - =- +

Step 4 Subtract. The remainder is 0.

Now Try:

5. Divide 24 5 6

2x x

x- --

.

_____________

2

2

2 5

3 2 11 15

2 6 5 15 5 15 0

x

x x x

x xxx

-- - +

-- +- +

22 11 15 2 5

3x x x

x- + = --

Check ( )( ) 2

2

3 2 5 2 5 6 15

2 11 15

x x x x x

x x

- - = - - +

= - +

6. Divide 3 28 9 7 9

3 1x x x

x+ - -

-.

Write the dividend in descending powers as 3 29 9 8 7x x x- + - .

Step 1 39x divided by 3x is 23 .x ( )2 3 23 3 1 9 3x x x x- = -


Step 3 26x- divided by 3x is –2x. ( ) 22 3 1 6 2x x x x- - =- +


Step 5 6x divided by 3x is 2. ( )2 3 1 6 2x x- = -

6. Divide 2 312 10 3 8

5 1x x x

x- + - -

-.

_____________



2

3 2

3 2

2

2

3 2 2

3 1 9 9 8 7

9 3

6 8

6 2 6 7 6 2 5

x x

x x x x

x x

x x

x xxx

- +- - + -

-- +- +

---

3 2

29 9 8 7 53 2 23 1 3 1

x x x x xx x

- + - -= - + +- -

Step 7 Multiply to check.

Check ( )( )( )( ) ( )( )

( )( ) ( )( )

2

2

3 2 2

3 2

53 1 3 2 23 1

3 1 3 3 1 2

5 3 1 2 3 13 1

9 3 6 2 6 2 5

9 9 8 7

x x xx

x x x x

x xx

x x x x x

x x x

-- - + +-

= - + - -

-+ - + --

= - - + + - -

= - + -

7. Divide 3 64x - by 4x- .

Here the dividend is missing the 2-termx and the -termx . We use 0 as the coefficient for each missing term.

2

3 2

3 2

2

2

4 16

4 0 0 64

4

4 0

4 16 16 64 16 64 0

x x

x x x x

x x

x x

x xxx

+ +- + + -

-+-

--

The remainder is 0. The quotient is 2 4 16.x x+ +

Check ( )( )2

3 2 2

3

4 4 16

4 16 4 16 64

64

x x x

x x x x x

x

- + +

= + + - - -

= -

7. Divide 3 1000x - by 10x- .

_____________



8. Divide 4 3 23 7 8 14x x x x- + - + by 2 2x + .

Since 2 2x + is missing the x-term, we write it

as 2 0 2x x+ + .

2

2 4 3 2

4 3 2

3 2

3 2

2

2

3 5

0 2 3 7 8 14

0 2

3 5 8

3 0 6

5 2 14

5 0 10 2 4

x x

x x x x x x

x x x

x x x

x x x

x x

x xx

- ++ + - + - +

+ +- + -- + -

- ++ +- +

The quotient is 222 43 5

2xx x

x- +- + +

+

The check shows that the quotient multiplied by the divisor gives the original dividend.

8. Divide 4 3 23 5 7 12 9x x x x+ - - + by

2 4x - .

_____________



Perform each division.

4. 26 23 202 5

x xx

- + --

4. _________________

5. 4 3 2

26 12 13 5 1

2 3x x x x

x- + - -

+ 5. _________________



6. 4 2

22 5 3

2 3a a

a+ +

+ 6. _________________

Objective 3 Use division in a geometry problem.

Video Examples

Review this example for Objective 3: 10. The area of a rectangle is given by

3 212 7 5 1p p p- + - square units, and the width is 4 1p- units. What is the length of the rectangle?

For a rectangle, A LW= . Solving for L gives AL

W= . Divide the area, 3 212 7 5 1p p p- + -

by the width 4 1p- .

Now Try: 10. The area of a rectangle is given

by 3 26 5 16 5r r r- + - square units, and the width is 3 1r- units. What is the length of the rectangle?

_____________

2

3 2

3 2

2

2

3 1

4 1 12 7 5 1

12 3

4 5

4 4 1 4 1 0

p p

p p p p

p p

p p

p ppp

- +

- - + --

- +- +

--

The length is 23 1p p- + units.





Work each problem.

7. The area of a parallelogram is given by 34 44 600y y- - square units, and the height is 6y- units. What is the base of the parallelogram?

7. _________________

8. The area of a parallelogram is given by 3 23 16 32 64t t t+ - - square units, and the base is

2 4 16t t+ - units. What is the height of the parallelogram?

8. _________________

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