Polynomials: Operations

4
4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Integers as Exponents Exponents and Scientific Notation Introduction to Polynomials Addition and Subtraction of Polynomials Multiplication of Polynomials Special Products Operations with Polynomials in Several Variables Division of Polynomials

Real-World Application
Total revenue of NASCAR (National Association of Stock Car Automobile Racing) is expected to be $3423 million by 2006. Convert this number to scientific notation.
Source: NASCAR

This problem appears as Exercise 64 in Section 4.2.

ISBN:0-536-47742-6

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Objectives
Tell the meaning of exponential notation. Evaluate exponential expressions with exponents of 0 and 1. Evaluate algebraic expressions containing exponents. Use the product rule to multiply exponential expressions with like bases. Use the quotient rule to divide exponential expressions with like bases. Express an exponential expression involving negative exponents with positive exponents.

4.1 a a a a

INTEGERS AS EXPONENTS

Exponential Notation
An exponent of 2 or greater tells how many times the base is used as a factor. For example, a 4.

In this case, the exponent is 4 and the base is a. An expression for a power is called exponential notation. an This is the exponent.

This is the base.
EXAMPLE 1

What is the meaning of 35 ? of n 4 ? of 2n 3 ? of 50x 2? of n 4 means n n n n; 50x 2 means 50 x x; n; n 3 means 1 n n n

n 3?

of

n ? 35 means 3 3 3 3 3; 2n 3 means 2n 2n 2n; n 3 means n n

3

Do Exercises 1–6. What is the meaning of each of the following? 1. 54 2. x 5

We read exponential notation as follows: a n is read the nth power of a, or simply a to the nth, or a to the n. We often read x 2 as “x-squared.” The reason for this is that the area of a square of side x is x x, or x 2. We often read x 3 as “x-cubed.” The reason for this is that the volume of a cube with length, width, and height x is x x x, or x 3.

x

x x

3. 3t

2

4. 3t

2

x

x

One and Zero as Exponents
Look for a pattern in the following:
5. x
4

6.

y

3

On each side, we divide by 8 at each step.

8 8 8 8 8 8 8 8 8 8 1

84 8
3

82 8? 8?.

On this side, the exponents decrease by 1.

ISBN:0-536-47742-6

To continue the pattern, we would say that 8 81 80.

Answers on page A-15 222
CHAPTER 4: Polynomials: Operations

and

1

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

We make the following definition.
EXPONENTS OF 0 AND 1

Evaluate. 7. 61 8. 70

a1 a
0

a, for any number a; 1, for any nonzero number a

We consider 0 0 to be not defined. We will explain why later in this section.
EXAMPLE 2

Evaluate 51, 8 1;
1 0

8 1, 30, 1;
0

7.3 0, and 186,892,046 0. 1

9. 8.4

1

10. 86540

51

5; 7.3

8; 30 186,892,046

Do Exercises 7–12.

Evaluating Algebraic Expressions
Algebraic expressions can involve exponential notation. For example, the following are algebraic expressions: x 4, 3x
3

11.

1.4

1

12. 0 1

2,

a2

2ab

b 2.

We evaluate algebraic expressions by replacing variables with numbers and following the rules for order of operations.
EXAMPLE 3

Evaluate 1000 x4 1000 1000 1000 375 54 625

x 4 when x

5.

Answers on page A-15

1000

Substituting

5 5 5 5

Study Tips
AUDIO RECORDINGS
Your instructor can request a complete set of audio recordings designed to help lead you through each section of this textbook. If you have difficulty reading or if you want extra review, these recordings explain solution steps for examples, caution you about errors, give instructions to work margin exercises, and then review the solutions to the margin exercises. These recordings are ideal for use outside of class. To obtain these audio recordings, consult with your instructor and refer to the Preface of this text. Recordings are available as MP3 files within MyMathLab.

EXAMPLE 4

Area of a Compact Disc. The standard compact disc used for software and music has a radius of 6 cm. Find the area of such a CD (ignoring the hole in the middle). A r2 6 cm 3.14
2

r

6 cm

6 cm 6 cm 36 cm2 113.04 cm2 In Example 4, “cm2” means “square centimeters” and “ ” means “is approximately equal to.”
EXAMPLE 5
ISBN:0-536-47742-6

Evaluate 5x 3 when x

2.

When we evaluate with a negative number, we often use extra parentheses to show the substitution. 5x
3

5 10 10 1000

2
3

3

Substituting Multiplying within brackets first

10

10
Evaluating the power

223
4.1 Integers as Exponents

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

13. Evaluate t 3 when t

5.

EXAMPLE 6

Evaluate 5x 3 when x
3

2.

5x 3
14. Evaluate 5x when x
5

5 5 5 8 40

2 2

Substituting

2

2

Evaluating the power first 2 2 2 8

2.

15. Find the area of a circle when r 32 cm. Use 3.14 for .

Recall that two expressions are equivalent if they have the same value for all meaningful replacements. Note that Examples 5 and 6 show that 5x 3 and 5x 3 are not equivalent—that is, 5x 3 5x 3.
Do Exercises 13–18.

16. Evaluate 200

a 4 when a

3.

Multiplying Powers with Like Bases
17. Evaluate t 1 when t 7. 4 and t 0 4

There are several rules for manipulating exponential notation to obtain equivalent expressions. We first consider multiplying powers with like bases: a3 a 2 a a a a a
⎫ ⎪ ⎬ ⎪ ⎭

a a a a a
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

a 5.

⎫ ⎬ ⎭

18. a) Evaluate 4t 2 when t

3.

3 factors

2 factors

5 factors

b) Evaluate 4t 2 when t

3.

Since an integer exponent greater than 1 tells how many times we use a base as a factor, then a a a a a a a a a a a 5 by the associative 5 law. Note that the exponent in a is the sum of those in a 3 a 2, that is, 3 2 5. Likewise, b4 b 3 b b b b b b b b 7, where 4 3 7.

c) Determine whether 4t 2 and 4t 2 are equivalent.

Adding the exponents gives the correct result.
THE PRODUCT RULE

For any number a and any positive integers m and n,
Multiply and simplify. 19. 3
5

am an

am

n

.

3

5

(When multiplying with exponential notation, if the bases are the same, keep the base and add the exponents.)

20. x 4 x 6

EXAMPLES Multiply and simplify. By simplify, we mean write the expression as one number to a nonnegative power.

21. p 4p 12p 8

7. 56 52 8. x 3 x 9

56 58 x3

2

Adding exponents: a m a n

am

n

9

22. x x 4

x 12
10 3

9. m 5m 10m 3 10. x x 8 x1 x9 11. a 3b 2 a 3b 5

m5
8

m 18

x1 x 8

Writing x as x 1
ISBN:0-536-47742-6

23. a 2b 3 a 7b 5

a 3a 3 b 2b 5 a 6b 7

Answers on page A-15 224
CHAPTER 4: Polynomials: Operations

Do Exercises 19–23.

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Dividing Powers with Like Bases
The following suggests a rule for dividing powers with like bases, such as a 5 a 2: a5 a2 a a a a a a a a a a a 3. a 2, that is, a a a a a 1 a a a a a a a 1 a a a a a 1 1

Divide and simplify. 24. 45 42

Note that the exponent in a 3 is the difference of those in a 5 5 2 3. In a similar way, we have t9 t4 t t t t t t t t t t t t t t 5, where 9 4 5.

Subtracting exponents gives the correct answer.
25.
THE QUOTIENT RULE

y6 y2

For any nonzero number a and any positive integers m and n, am an am n .

(When dividing with exponential notation, if the bases are the same, keep the base and subtract the exponent of the denominator from the exponent of the numerator.)

EXAMPLES Divide and simplify. By simplify, we mean write the expression as one number to a nonnegative power.

26.

p 10 p

12.

65 63 t 12 t

65 62

3

Subtracting exponents

13.

x8 x2 p 5q 7 p 2q 5

x8 x6

2

14.

t 12 t1

t 12 t 11

1

15.

p 5 q7 p2 q5

p 5 2q 7 p 3q 2

5

The quotient rule can also be used to explain the definition of 0 as an exponent. Consider the expression a 4 a 4, where a is nonzero: a4 a4 a a a a a a a a
27.

1.

a 7b 6 a 3b 4

This is true because the numerator and the denominator are the same. Now suppose we apply the rule for dividing powers with the same base: a4 a4
ISBN:0-536-47742-6

a4

4

a 0.

Since a 4 a 4 1 and a 4 a 4 a 0, it follows that a 0 1, when a 0. We can explain why we do not define 0 0 using the quotient rule. We know that 0 0 is 0 1 1. But 0 1 1 is also equal to 0 1 0 1, or 0 0. We have already seen that division by 0 is not defined, so 0 0 is also not defined.
Do Exercises 24–27.

Answers on page A-15 225
4.1 Integers as Exponents

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Express with positive exponents. Then simplify. 28. 4
3

Negative Integers as Exponents
We can use the rule for dividing powers with like bases to lead us to a definition of exponential notation when the exponent is a negative integer. Consider 53 57 and first simplify it using procedures we have learned for working with fractions: 53 57 5 5 5 5 5 5 5 5 5 5 5 5 5 1 5 5 5 5 5 5 5 5 5 5 1 5 5 5 5 5 5 5 1 . 54

29. 5

2

Now we apply the rule for dividing exponential expressions with the same bases. Then 53 57
4

53

7

5 4.

From these two expressions for 53 57, it follows that
30. 2

5

4

1 . 54

This leads to our definition of negative exponents.
NEGATIVE EXPONENT

31.

2

3

For any real number a that is nonzero and any integer n, a n 1 . an

In fact, the numbers a n and a an a
32. 4p
3

n

are reciprocals of each other because

n

an

1 an

an an

1.

The following is another way to arrive at the definition of negative exponents. On each side, we divide by 5 at each step. 5 5 5 5 5 5 5 5 5 5
33. 1 x
2

54 5
3

52 51 50 5? 5?

On this side, the exponents decrease by 1.

1 1 5 1 25

ISBN:0-536-47742-6

To continue the pattern, it should follow that 1 5 1 51 5
1

Answers on page A-15 226
CHAPTER 4: Polynomials: Operations

and

1 25

1 52

5 2.

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

EXAMPLES

Express using positive exponents. Then simplify. 1 16 17. 19. ab
3

Simplify.

16. 4 18. m 20. 1 x

2

1 42
3

3
1

2

1 3 1 a 1 b 3 1 c5

1
2

3 1 a b 3 c5

3 a b

1 9

34. 5

2

54

1 m3 x x3

3

21. 3c

5

Example 20 might also be done as follows: 1 x
3

1 1 x3

1

x3 1

x 3.

35. x

3

x

4

Caution!
As shown in Examples 16 and 17, a negative exponent does not necessarily mean that an expression is negative. Do Exercises 28–33 on the preceding page.

The rules for multiplying and dividing powers with like bases still hold when exponents are 0 or negative. We state them in a summary below.
EXAMPLES Simplify. By simplify, we generally mean write the expression as one number or variable to a nonnegative power.

36.

7 2 73

22. 7

3

76

7 7
3 2 2 4 4 5

3 6

Adding exponents Subtracting exponents

23. x 4 x x x7

3

x4
7

3

x1 1 x6

x

24.

54 5 2
4 5

54 54

25.

x1

x

6

56
5

37.

b 26. b

b b

2 3

b b

27. y b1 b

4

y

8

y y

4 12

8

1 y 12

Do Exercises 34–38.

The following is a summary of the definitions and rules for exponents that we have considered in this section.
DEFINITIONS AND RULES FOR EXPONENTS

38.

t t
5

1 as an exponent: 0 as an exponent:

a1 a0 n a 1, a 0 a n; a n Negative integers as exponents: a
ISBN:0-536-47742-6

1 1 , an a n am n 0

Product Rule: Quotient Rule:

am an a an m am

,a

0

Answers on page A-15 227
4.1 Integers as Exponents
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.1
1. 34 7. 7p
2

EXERCISE SET

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Student’s Solutions Manual

What is the meaning of each of the following? 2. 43 3. 1.1
5

4. 87.2

6

5.

2 3

4

6.

5 8

3

8. 11c

3

9. 8k 3

10. 17x 2

11.

6y 4

12.

q5

Evaluate. 13. a 0, a 0 14. t 0, t 0 15. b 1 16. c 1 17. 2 3
0

18.

5 8

0

19.

7.03

1

20.

4 5

1

21. 8.380

22. 8.381

23. ab

1

24. ab 0, a, b

0

25. ab 0

26. ab 1

Evaluate. 27. m 3, when m 3 28. x 6, when x 2 29. p 1, when p 19 30. x 19, when x 0

31.

x 4, when x

3

32.

2y 7, when x

2

33. x 4, when x

4

34. y 15, when y

1

35. y 2

7, when y

10

36. z 5

5, when z

2

37. 161 b 2, when b 5

38. 325 v

v 3, when 3

39. x 1

3 and x 0

3, when x

7

40. y 0

8 and y 1

8, when y

3

41. Find the area of a circle when r r 34 ft. Use 3.14 for .

42. The area A of a square with sides of length s is given by A s 2. Find the area of a square with sides of length 24 m.

s
Copyright © 2007 Pearson Education, Inc.

s

ISBN:0-536-47742-6

Express using positive exponents. Then simplify. 43. 3
2

44. 2

3

45. 10

3

46. 5

4

47. 7

3

228
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

48. 5

2

49. a

3

50. x

2

51.

1 8 2

52.

1 2 5

53.

1 y
4

54.

1 t
7

55.

1 z n 56.

1 h n

Express using negative exponents. 57. 1 43 58. 1 52 59. 1 x3 60. 1 y2 61. 1 a5 62. 1 b7

, 63. 24 23

Multiply and simplify. 64. 35 32 65. 85 89 66. n 3 n 20

67. x 4 x 3

68. y 7 y 9

69. 9 17 9 21

70. t 0 t 16

71. 3y

4

3y

8

72. 2t

8

2t

17

73. 7y

1

7y

16

74. 8x

0

8x

1

75. 3

5

38

76. 5

8

59

77. x

2

x

78. x x

1

79. x 14 x 3

80. x 9 x 4

81. x

7

x

6

82. y

5

y

8

83. a 11 a
ISBN:0-536-47742-6

3

a

18

84. a

11

a

3

a

7

85. t 8 t

8

86. m 10 m

10

229
Exercise Set 4.1
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

, 87. 75 72

Divide and simplify. 88. 58 56 89. 812 86 90. 813 82

91.

y9 y5

92.

x 11 x9

93.

162 168

94.

72 79

95.

m6 m 12

96.

a3 a4

97.

8x 6 8x 10

98.

8t 4 8t 11

99.

2y 2y

9 9

100.

6y 6y

7 7

101.

x x
1

102.

y8 y

103.

x7 x 2

104.

t8 t 3

105.

z z

6 2

106.

x x

9 3

107.

x x

5 8

108.

y y

2 9

109.

m m

9 9

110.

x x

7 7

Matching. In Exercises 111 and 112, match each item in the first column with the appropriate item in the second column by drawing connecting lines. 111. 52 5 1 5 1 5 52 5 1 5 1 5
2 2 2

1 10 1 10
2

112.

1 8 1 8 8

2

16 16
2

64
2

1 25 10 25 25

64 1 64 1 64

82 82 8 1 8 1 8
2 2 2

2

1 25
2

1 16 1 16

10

Copyright © 2007 Pearson Education, Inc.

ISBN:0-536-47742-6

230
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

113.

DW Suppose that the width of a square is three times the width of a second square. How do the areas of the squares compare? Why?

114.

DW Suppose that the width of a cube is twice the

width of a second cube. How do the volumes of the cubes compare? Why?

3x 2x x x

SKILL MAINTENANCE
Solve. [2.6a] 115. Cutting a Submarine Sandwich. A 12-in. submarine sandwich is cut into two pieces. One piece is twice as long as the other. How long are the pieces? 117. The perimeter of a rectangle is 640 ft. The length is 15 ft more than the width. Find the area of the rectangle. Solve. [2.3c] 119. 62 x 10 5x 7 10 120. 10 x 4 5 2x 5 7 116. Book Pages. A book is opened. The sum of the page numbers on the facing pages is 457. Find the page numbers. 118. The first angle of a triangle is 24° more than the second. The third angle is twice the first. Find the measures of the angles of the triangle.

Factor. [1.7d] 121. 4x 12 24y 122. 256 2a 4b

SYNTHESIS
Determine whether each of the following is correct. 123. x Simplify. 127. y 2x
1 4 2 1 5 2

1

2

x2

1

124. x

1

2

x2

2x

1

125. 5x

0

5x 0

126.

x3 x5

x2

y 3x

128. a 5k

a 3k

129.

a 6t a 7t a 9t

130.

131. b 2 and a 2

0.8 5 0.8 3 0.8

2

132. Determine whether a Use
ISBN:0-536-47742-6

b 2 are equivalent. (Hint: Choose values for a and b and evaluate.)

,

, or 34

for

to write a true sentence. 134. 42 43 135. 43 53 136. 43 34

133. 35 Evaluate. 137.

1 , when z z4

10

138.

1 , when z z5

0.1

231
Exercise Set 4.1
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Objectives
Use the power rule to raise powers to powers. Raise a product to a power and a quotient to a power. Convert between scientific notation and decimal notation. Multiply and divide using scientific notation. Solve applied problems using scientific notation.

4.2

EXPONENTS AND SCIENTIFIC NOTATION

We now enhance our ability to manipulate exponential expressions by considering three more rules. The rules are also applied to a new way to name numbers called scientific notation.

Raising Powers to Powers
Consider an expression like 32 4. We are raising 32 to the fourth power: 32
4

32 32 32 32 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 38.

Simplify. Express the answers using positive exponents. 1. 34
5

Note that in this case we could have multiplied the exponents: 32
4

32 4

38. y 24. Once again, we get the same result if

y8 y8 y8 Likewise, y 8 3 we multiply the exponents: y8
3

y8 3

y 24.

THE POWER RULE

For any real number a and any integers m and n,
2. x
3 4

am

n

a mn.

(To raise a power to a power, multiply the exponents.)

EXAMPLES

Simplify. Express the answers using positive exponents.
20

1. 35
5 3

4

35 4 3

Multiplying exponents
57

2. 22

5

22 5

210 1 x8

3.

y

3.

y

5 7 4 6

y a

y
6

35

1 y 35 a 24

4. x 4

2

x4

2

x

8

5. a

4

Do Exercises 1–4.

Raising a Product or a Quotient to a Power
4. x 4
8

When an expression inside parentheses is raised to a power, the inside expression is the base. Let’s compare 2a 3 and 2a 3: 2a 3 2a
3

2 a a a; 2a 2a 2a

The base is a. The base is 2a. Using the associative and commutative laws of multiplication to regroup the factors
ISBN:0-536-47742-6

2 2 2 a a a

Answers on page A-16 232
CHAPTER 4: Polynomials: Operations

23a 3 8a 3.

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

We see that 2a 3 and 2a 3 are not equivalent. We also see that we can evaluate the power 2a 3 by raising each factor to the power 3. This leads us to the following rule for raising a product to a power.

Simplify. 5. 2x 5y
3 4

RAISING A PRODUCT TO A POWER

For any real numbers a and b and any integer n, ab n a nb n.

6. 5x 5y

6

z

3 2

(To raise a product to the nth power, raise each factor to the nth power.)

EXAMPLES

Simplify. 41x 2 41 4
3 3 3

7. Since 4 41
3 6

x

37 2

6. 4x 2

3

x2
6

Raising each factor to the third power

x

64x
4

7. 5x 3y 5z 2 8. 9. 5x 4y 3 x

4

54 x 3 5 x 1 1 x x 50
3

y5
3

4

z2
3

4

Raising each factor to the fourth power 8. 3y

625x y z
3

12 20 8

2

x

5 8 3

z

x4

y3

125x 12y 9
25 2 50 50

Using the power rule Using the property of 1 (Section 1.8) 9. The product of an even number of negative factors is positive. y8
3

1 x x
50

50 50

10. 5x 2y

2 3

53 x 2

3

y

2 3

125x 6y 125x 6 y6

6

Be sure to raise each factor to the third power. 10. 2x 4
3

11. 3x 3y 12. x4

5 2 4

z

34 x 3 1 x4 1 1
3

4 3

y

5 4

z2 1

4 3

81x 12y x4
3

20 8

z

81x 12z 8 y 20
3

3

1

x

12

1 x 12 2
3

1 1 1 x 12 x
5 3

1 x 12
3

11.

3x 2y

5

3

13.
ISBN:0-536-47742-6

2x

5 4

y

3

y4

1 2

3

x 15 y

12

1 1 x 15 12 8 y
Do Exercises 5–11.

x 15 8y 12

Answers on page A-16 233
4.2 Exponents and Scientific Notation

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Simplify. 12. x6 5
2

There is a similar rule for raising a quotient to a power.
RAISING A QUOTIENT TO A POWER

For any real numbers a and b, b a b n 0, and any integer n,

a . bn

n

(To raise a quotient to the nth power, raise both the numerator and the denominator to the nth power.) Also, a b n EXAMPLES

⎫ ⎪ ⎬ ⎪ ⎭ b a n bn , a an

0.

Simplify. x2 3 43 x6 64 32 a 4 b3 2
2 2 2

13.

2t 5 w4

3

14. 15.

x 4

2 3

3a 4 b3 y3 5

2

3a 4 2 b3 2 y3 5

9a 8 b6 1 y6 1 52 1 y6 1 52 1 52 y6 1 25 y6

2

16.

y 5

6 2

Example 16 might also be done as follows: y3 5
2

5 y3 52 y3 2

2

a b

n

b a

n

25 . y6

14.

x4 2 3 Do this two ways.

Do Exercises 12–14.

Scientific Notation
There are many kinds of symbols, or notation, for numbers. You are already familiar with fraction notation, decimal notation, and percent notation. Now we study another, scientific notation, which makes use of exponential notation. Scientific notation is especially useful when calculations involve very large or very small numbers. The following are examples of scientific notation.

ISBN:0-536-47742-6

Answers on page A-16 234
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

1

2

3

1 Niagara Falls: On the Canadian side, during the summer the amount of water that spills over the falls in 1 day is about 4.9793 10 10 gal 49,793,000,000 gal. 10 21 6,615,000,000,000,000,000,000 tons.

2 The mass of the earth: 6.615 3 The mass of a hydrogen atom: 1.7 10
24

g

0.0000000000000000000000017 g.

SCIENTIFIC NOTATION

Scientific notation for a number is an expression of the type M 10 n,

where n is an integer, M is greater than or equal to 1 and less than 10 1 M 10 , and M is expressed in decimal notation. 10 n is also considered to be scientific notation when M 1.

You should try to make conversions to scientific notation mentally as much as possible. Here is a handy mental device. A positive exponent in scientific notation indicates a large number (greater than or equal to 10) and a negative exponent indicates a small number (between 0 and 1).

EXAMPLES

Convert to scientific notation. 7.8 10
4

Caution!
Each of the following is not scientific notation. 12.46 10 7
⎫ ⎬ ⎭

17. 78,000

7.8,000. 4 places Large number, so the exponent is positive. 18. 0.0000057
ISBN:0-536-47742-6

5.7

10

6

This number is greater than 10. 0.347 10
5

0.000005.7 6 places Small number, so the exponent is negative.
Do Exercises 15 and 16 on the following page.

This number is less than 1.

4.2 Exponents and Scientific Notation
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

⎫ ⎬ ⎭

235

Convert to scientific notation. 15. 0.000517

EXAMPLES

Convert mentally to decimal notation. 10 5 789,300

19. 7.893

7.89300. 5 places Positive exponent, so the answer is a large number. 20. 4.7
16. 523,000,000

10

8

0.000000047 8 places

.00000004.7 Negative exponent, so the answer is a small number.
Do Exercises 17 and 18.

Convert to decimal notation. 17. 6.893 10 11

Multiplying and Dividing Using Scientific Notation
MULTIPLYING
Consider the product 400 2000 800,000.

In scientific notation, this is 4
18. 5.67 10
5

10 2

2

10 3

4 2 10 2 10 3

8

10 5.

By applying the commutative and associative laws, we can find this product by multiplying 4 2, to get 8, and 10 2 10 3, to get 10 5 (we do this by adding the exponents).
EXAMPLE 21

Multiply: 1.8 2.3 10
4

10 6

2.3

10

4

.
4

We apply the commutative and associative laws to get 1.8
Multiply and write scientific notation for the result. 19. 1.12 10
8

10 6

1.8 2.3 4.14 4.14 10 6 10 2.

10 6 10
4

5

10

7

We get 4.14 by multiplying 1.8 and 2.3. We get 10 2 by adding the exponents 6 and 4.
EXAMPLE 22

Multiply: 3.1 10
3

10 5

4.5

10
3

3

.

We have 3.1
20. 9.1 10
17

10 5

4.5

3.1 13.95 1.395 1.395 1.395

4.5 10 5 10 10
2

8.2

10 3

Not scientific notation. 13.95 is greater than 10.

10 1 10 1 10 .
3

10 2 10 2

Substituting 1.395 for 13.95 Associative law

10 1

ISBN:0-536-47742-6

Adding exponents. The answer is now in scientific notation.

Answers on page A-16 236
CHAPTER 4: Polynomials: Operations

Do Exercises 19 and 20.

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

DIVIDING
Consider the quotient 800,000 400 2000.

Divide and write scientific notation for the result. 21. 4.2 2.1 10 5 10 2

In scientific notation, this is 8 10 5 4 10 2 8 4 10 5 10 2 8 4 10 5 10 2 2 10 3.

We can find this product by dividing 8 by 4, to get 2, and 10 5 by 10 2, to get 10 3 (we do this by subtracting the exponents.)
22. EXAMPLE 23

Divide: 3.41

10 5

1.1

10

3

. 10 5 10 3

1.1 2.0

10 10

4 7

We have 3.41 10 5 1.1 10
3

3.41 10 5 1.1 10 3 3.1 3.1 10 5 10 8.
3

3.41 1.1

Answers on page A-16

CALCULATOR CORNER
1.789E 11 1.789E 11

NORMAL SCI ENG FLOAT 0123456789 RADIAN DEGREE FUNC PAR POL SEQ CONNECTED DOT SEQUENTIAL SIMUL REAL a bi re^ i FULL HORIZ G T

Scientific Notation To enter a number in scientific notation on a graphing calculator, we first type the decimal portion of the number and then press F $. (EE is the second operation associated with the , key.) Finally, we type the exponent, which can be at most two digits. For example, to enter 1.789 10 11 in scientific notation, we press 1 . 7 8 9 F $ : 1 1 [. The decimal portion of the number appears before a small E and the exponent follows the E. The graphing calculator can be used to perform computations using scientific notation. To find the product in Example 21 and express the result in scientific notation, we first set the calculator in Scientific mode by pressing G, positioning the cursor over Sci on the first line, and pressing [. Then we press F o to go to the home screen and enter the computation by pressing 1 . 8 F $ 6 b 2 . 3 F $ : 4 [. Exercises: Multiply or divide and express the answer in scientific notation.
1. 3.15 2. 4.76 10 7 4.3 10 10 10
9 4 5

1.8E6 2.3E 4 4.14E2

10 10
5

12 10

1.9 10 10 7

On a scientific calculator, the answer may appear as follows.

3. 8 4. 4 5. 4.5 1.5 6.4 1.6 4 5 9 3

4 9

10 6 10 12 10 10 10 9 10 16 10 11 10 2
5 10

4.14 02

6. 7. 8.

ISBN:0-536-47742-6

237
4.2 Exponents and Scientific Notation
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

23. Niagara Falls Water Flow. On the Canadian side, during the summer the amount of water that spills over the falls in 1 min is about 1.3088 10 8 L.

EXAMPLE 24

Divide: 6.4

10 6.4 8.0 6.4 8.0 0.8 0.8 8.0 8.0 8.0

7

8.0

10 6 .

We have 6.4 10
7

8.0

10 6

10 7 10 6 10 7 10 6 10 10 10 10 10
7 6 13

How much water spills over the falls in one day? Express the answer in scientific notation.

Not scientific notation. 0.8 is less than 1.

1

10 10 .

13

Substituting 8.0 10 1 for 0.8 Associative law Adding exponents

1 14

13

Do Exercises 21 and 22 on the preceding page.

Applications with Scientific Notation
EXAMPLE 25

Distance from the Sun to Earth. Light from the sun traveling at a rate of 300,000 kilometers per second (km s) reaches Earth in 499 sec. Find the distance, expressed in scientific notation, from the sun to Earth.

Answer on page A-16

Study Tips
?

WRITING ALL THE STEPS
Take the time to include all the steps when working your homework problems. Doing so will help you organize your thinking and avoid computational errors. If you find a wrong answer, having all the steps allows easier checking of your work. It will also give you complete, stepby-step solutions of the exercises that can be used to study for an exam. Writing down all the steps and keeping your work organized may also give you a better chance of getting partial credit. “Success comes before work only in the dictionary.”
Anonymous

The time t that it takes for light to reach Earth from the sun is 4.99 10 2 sec (s). The speed is 3.0 10 5 km s. Recall that distance can be expressed in terms of speed and time as Distance d We substitute 3.0 d rt 3.0 14.97 1.497 1.497 1.497 10 5 4.99 10
7

Speed Time rt. 10 5 for r and 4.99 10 2 10 7 10 7
Converting to scientific notation

10 2 for t:

Substituting

10 1 10
1

ISBN:0-536-47742-6

10 8 km.

Thus the distance from the sun to Earth is 1.497
Do Exercise 23.

10 8 km.

238
CHAPTER 4: Polynomials: Operations

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

DNA. A strand of DNA (deoxyribonucleic acid) is about 150 cm long and 1.3 10 10 cm wide. How many times longer is DNA than it is wide?
EXAMPLE 26
Source: Human Genome Project Information

24. Earth vs. Saturn. The mass of Earth is about 6 10 21 metric tons. The mass of Saturn is about 5.7 10 23 metric tons. About how many times the mass of Earth is the mass of Saturn? Express the answer in scientific notation.

To determine how many times longer (N ) DNA is than it is wide, we divide the length by the width: N 1.3 150 10
10

150 1.3

1 10 10 10 10 10 2 10 12. 10 12 times its width. 10 10

115.385 1.15385 1.15385

Thus the length of DNA is about 1.15385
Do Exercise 24.

The following is a summary of the definitions and rules for exponents that we have considered in this section and the preceding one.

DEFINITIONS AND RULES FOR EXPONENTS

Exponent of 1: Exponent of 0: Negative exponents: Product Rule: Quotient Rule: Power Rule: Raising a product to a power: Raising a quotient to a power:
ISBN:0-536-47742-6

a1 a0 a n a 1, a 0 a n, a n 1 1 , an a n am n 0

am an a an m am n n n

,a

0

am ab a b a b

a mn a nb n an ,b bn n 0; 0, a 0 M 10

bn ,b an

Scientific notation:

M

10 n, or 10 n, where 1

Answer on page A-16 239
4.2 Exponents and Scientific Notation
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.2
, 1. 23
2

EXERCISE SET

For Extra Help

MathXL

MyMathLab

InterAct Math

Math Tutor Digital Video Center Tutor CD 2 Videotape 4

Student’s Solutions Manual

Simplify. 2. 52
4

3. 52

3

4. 7

3 5

5. x

3

4

6. a

5

6

7. a

2 9

8. x

5 6

9. t

3

6

10. a

4

7

11. t 4

3

12. t 5

2

13. x

2

4

14. t

6

5

15. ab

3

16. xy

2

17. ab

3

18. xy

6

19. mn 2

3

20. x 3y

2

21. 4x 3

2

22. 4 x 3

2

23. 3x

4 2

24. 2a

5 3

25. x 4y 5

3

26. t 5x 3

4

27. x

6

y

2

4

28. x

2

y

7

5

29. a

2 7

b

5

30. q 5r

1

3

31. 5r

4 3 2

t

32. 4x 5y

6 3

33. a

5 7

b c

2 3

34. x

4

y

2 9 2

z

35. 3x 3y

8

z

3 2

36. 2a 2y

4

z

5 3

37.

4x 3y

2 2

38.

8x 3y

2 3

39.

a

3

b

2

4

40.

p

4

q

3

2

41.

y3 2

2

42.

a5 3

3

43.

a2 b3

4

44.

x3 y4

5

45.

y2 2

3

46.

a4 3

2

47.

7 x
3

2

48.

3 a 2

3

49.

x 2y z

3

50.

m n 4p

3

51.

a 2b cd 3

2

52.

2a 2 3b 4

3

Copyright © 2007 Pearson Education, Inc.

ISBN:0-536-47742-6

240
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Convert to scientific notation. 53. 28,000,000,000 54. 4,900,000,000,000 55. 907,000,000,000,000,000 56. 168,000,000,000,000

57. 0.00000304

58. 0.000000000865

59. 0.000000018

60. 0.00000000002

61. 100,000,000,000

62. 0.0000001

Convert the number in the sentence to scientific notation. 63. Population of the United States. As of July 2005, the population of the United States was about 296 million (1 million 10 6).
Source: U.S. Bureau of the Census

64. NASCAR. Total revenue of NASCAR (National Association of Stock Car Automobile Racing) is expected to be $3423 million by 2006.
Source: NASCAR

65. State Lottery. Typically, the probability of winning a state lottery is about 1 10,000,000.

66. Cancer Death Rate. In Michigan, the death rate due to cancer is about 127.1 1000.
Source: AARP

Convert to decimal notation. 67. 8.74 10 7 68. 1.85 10 8 69. 5.704 10
8

70. 8.043

10

4

71. 10 7

72. 10 6

73. 10

5

74. 10

8

Multiply or divide and write scientific notation for the result. 75. 3 10 4 2 10 5 76. 3.9 10 8 8.4 10
3

77. 5.2

10 5 6.5

10

2

ISBN:0-536-47742-6

78. 7.1

10

7

8.6

10

5

79. 9.9

10

6

8.23

10

8

80. 1.123

10 4

10

9

241
Exercise Set 4.2
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

81.

8.5 3.4

10 8 10 5

82.

5.6 2.5

10 2 10 5

83. 3.0

10 6

6.0

10 9

84. 1.5

10

3

1.6

10

6

85.

7.5 2.5

10 9 10 12

86.

4.0 8.0

10 3 10 20

Solve. 87. River Discharge. The average discharge at the mouths of the Amazon River is 4,200,000 cubic feet per second. How much water is discharged from the Amazon River in 1 yr? Express the answer in scientific notation. 88. Computers. A gigabyte is a measure of a computer’s storage capacity. One gigabyte holds about one billion bytes of information. If a firm’s computer network contains 2500 gigabytes of memory, how many bytes are in the network? Express the answer in scientific notation.

Brazil ve azon R i r Mouths of the Amazon River

Am

89. Earth vs. Jupiter. The mass of Earth is about 6 10 21 metric tons. The mass of Jupiter is about 1.908 10 24 metric tons. About how many times the mass of Earth is the mass of Jupiter? Express the answer in scientific notation.
Earth Jupiter

90. Water Contamination. In the United States, 200 million gal of used motor oil is improperly disposed of each year. One gallon of used oil can contaminate one million gallons of drinking water. How many gallons of drinking water can 200 million gallons of oil contaminate? Express the answer in scientific notation.
Source: The Macmillan Visual Almanac

Copyright © 2007 Pearson Education, Inc.

ISBN:0-536-47742-6

242
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

91. Stars. It is estimated that there are 10 billion trillion stars in the known universe. Express the number of stars in scientific notation (1 billion 10 9; 1 trillion 10 12).

92. Closest Star. Excluding the sun, the closest star to Earth is Proxima Centauri, which is 4.3 light-years away (one light-year 5.88 10 12 mi). How far, in miles, is Proxima Centauri from Earth? Express the answer in scientific notation.

93. Earth vs. Sun. The mass of Earth is about 6 10 21 metric tons. The mass of the sun is about 1.998 10 27 metric tons. About how many times the mass of Earth is the mass of the sun? Express the answer in scientific notation.

94. Red Light. The wavelength of light is given by the velocity divided by the frequency. The velocity of red light is 300,000,000 m sec, and its frequency is 400,000,000,000,000 cycles per second. What is the wavelength of red light? Express the answer in scientific notation.

Space Travel. Use the following information for Exercises 95 and 96.

APPROXIMATE DISTANCE FROM EARTH TO:
Moon Earth Mars

Moon Mars Pluto

240,000 miles 35,000,000 miles 2,670,000,000 miles

Pluto

95. Time to Reach Mars. Suppose that it takes about 3 days for a space vehicle to travel from Earth to the moon. About how long would it take the same space vehicle traveling at the same speed to reach Mars? Express the answer in scientific notation.

96. Time to Reach Pluto. Suppose that it takes about 3 days for a space vehicle to travel from Earth to the moon. About how long would it take the same space vehicle traveling at the same speed to reach Pluto? Express the answer in scientific notation.

97.
ISBN:0-536-47742-6

DW Explain in your own words when exponents

98.

should be added and when they should be multiplied.

DW Without performing actual computations, explain why 3
29

is smaller than 2

29

.

243
Exercise Set 4.2
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

SKILL MAINTENANCE
Factor. [1.7d] 99. 9x 36 100. 4x 2y 16 101. 3s 3t 24 102. 7x 14

Solve. [2.3b] 103. 2x 4 5x 8 x 3 104. 8x 7 9x 12 6x 5

Solve. [2.3c] 105. 8 2x 3 2x 5 10 106. 4 x 3 5 6x 2 8

Graph. [3.2a], [3.3a] 107. y x 5 y 5 4 3 2 1 5 4 3 2 1 1 2 3 4 5 1 2 3 4 5

108. 2x

y

8 y 5 4 3 2 1

x

5 4 3 2 1

1 2 3 4 5

1 2 3 4 5

x

SYNTHESIS
109. Carry out the indicated operations. Express the result in scientific notation. 5.2 10 6 6.1 1.28 10 10
3 11

110. Find the reciprocal and express it in scientific notation. 6.25 10
3

Simplify. 111. 512 2 525 112. a 22 a 2 11 113. 35
4

35 34

114.

5x 2 3y 2z

0

115.

49 18 735

116.

1 a

n

117.

0.4 5 0.43 2

118.

4a 3b 2 5c 3

1

Determine whether each of the following is true for all pairs of integers m and n and all positive numbers x and y. 119. x m y n xy mn 120. x m y m

xy

2m

121. x

y

m

xm

ym

Copyright © 2007 Pearson Education, Inc.

ISBN:0-536-47742-6

122.

xm

x

m

123.

x

2m

x 2m

124. x

m

1 xm

244
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.3
3x 2, 2x,

INTRODUCTION TO POLYNOMIALS

Objectives
Evaluate a polynomial for a given value of the variable. Identify the terms of a polynomial. Identify the like terms of a polynomial. Identify the coefficients of a polynomial. Collect the like terms of a polynomial. Arrange a polynomial in descending order, or collect the like terms and then arrange in descending order. Identify the degree of each term of a polynomial and the degree of the polynomial. Identify the missing terms of a polynomial. Classify a polynomial as a monomial, binomial, trinomial, or none of these.

We have already learned to evaluate and to manipulate certain kinds of algebraic expressions. We will now consider algebraic expressions called polynomials. The following are examples of monomials in one variable: 5, 37p 4, 0.

Each expression is a constant or a constant times some variable to a nonnegative integer power.
MONOMIAL

A monomial is an expression of the type ax n, where a is a realnumber constant and n is a nonnegative integer.

Algebraic expressions like the following are polynomials:
3 4

y ,

5

2, 5y

3, 3x

2

2x

5,

7a

3

1 2 a,

6x, 37p , x, 0.

4

POLYNOMIAL

A polynomial is a monomial or a combination of sums and/or differences of monomials.

The following algebraic expressions are not polynomials: (1) x x 3 , 4 (2) 5x 3 2x 2 1 , x (3) 1 x
3

2

.

1. Write three polynomials.

Expressions (1) and (3) are not polynomials because they represent quotients, not sums or differences. Expression (2) is not a polynomial because 1 x x
1

,

and this is not a monomial because the exponent is negative.
Do Exercise 1.

Evaluating Polynomials and Applications
When we replace the variable in a polynomial with a number, the polynomial then represents a number called a value of the polynomial. Finding that number, or value, is called evaluating the polynomial. We evaluate a polynomial using the rules for order of operations (Section 1.8).
EXAMPLE 1
ISBN:0-536-47742-6

Evaluate the polynomial when x b) 2x
2

2. 3 2 22 7 2 3 2 4 7 2 3 8 14 3 3

a) 3x

5

3 2 5 6 5 11

7x

Answer on page A-16 245
4.3 Introduction to Polynomials

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Evaluate the polynomial when x 3. 2. 4x 7

EXAMPLE 2

Evaluate the polynomial when x 4
3

4.

a) 2

x

3

2

2 64 2 64 66 4
2

3.

5x 3

7x

10

b)

x2

3x

1 16 3

3 1

4

1

12

Evaluate the polynomial when x 5. 4. 5x 7

Do Exercises 2–5.

AG ALGEBRAIC – GRAPHICAL CONNECTION
5. 2x 2 5x 4

6. Use only the graph shown in Example 3 to evaluate the polynomial 2x 2 when x 4 1. and when x

An equation like y 2x 2, which has a polynomial on one side and y on the other, is called a polynomial equation. Here and in many places throughout the book, we will connect graphs to related concepts. Recall from Chapter 3 that in order to plot points before graphing an equation, we choose values for x and compute the corresponding y-values. If the equation has y on one side and a polynomial involving x on the other, then determining y is the same as evaluating the polynomial. Once the graph of such an equation has been drawn, we can evaluate the polynomial for a given x-value by finding the y-value that is paired with it on the graph.
EXAMPLE 3

7. Referring to Example 4, determine the total number of games to be played in a league of 12 teams.

Use only the given graph of y 2x 2 to evaluate the polynomial 2x 2 when x 3. First, we locate 3 on the x-axis. From there we move vertically to the graph of the equation and then horizontally to the y-axis. There we locate the y-value that is paired with 3. Although our drawing may not be precise, it appears that the y-value 4 is paired with 3. Thus the value of 2x 2 is 4 when x 3.

y
5 4 3 2 1 5 4 3 2 1 1 2 1 2 3 4 5

8. Perimeter of a Baseball Diamond. The perimeter P of a square of side x is given by the polynomial equation P 4x.

x

y

2x

2

4 5

Do Exercise 6.

Polynomial equations can be used to model many real-world situations. x x

EXAMPLE 4 x x

Games in a Sports League. In a sports league of x teams in which each team plays every other team twice, the total number of games N to be played is given by the polynomial equation N x2 x.

A baseball diamond is a square 90 ft on a side. Find the perimeter of a baseball diamond.

A women’s slow-pitch softball league has 10 teams. What is the total number of games to be played? We evaluate the polynomial when x N x2 x 10 2 10 100 10 10:
ISBN:0-536-47742-6

90.

The league plays 90 games.

Answers on page A-16 246
CHAPTER 4: Polynomials: Operations

Do Exercises 7 and 8.

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

EXAMPLE 5

Medical Dosage. The concentration C, in parts per million, of a certain antibiotic in the bloodstream after t hours is given by the polynomial equation C 0.05t 2 2t 2.

9. Medical Dosage. a) Referring to Example 5, determine the concentration after 3 hr by evaluating the polynomial when t 3.

Find the concentration after 2 hr. t To find the concentration after 2 hr, we evaluate the polynomial when 2: C 0.05t 2 0.05 2 0.05 4 0.2 0.2 5.8. The concentration after 2 hr is 5.8 parts per million.
b) Use only the graph showing medical dosage to check the value found in part (a).
2

2t 22 22 2

2 2 2

Carrying out the calculation using the rules for order of operations

4 6

AG ALGEBRAIC – GRAPHICAL CONNECTION
The polynomial equation in Example 5 can be graphed if we evaluate the polynomial for several values of t. We list the values in a table and show the graph below. Note that the concentration peaks at the 20-hr mark and after slightly more than 40 hr, the concentration is 0. Since neither time nor concentration can be negative, our graph uses only the first quadrant.
C t 0 2 10 20 30 C 0.05t 2 2 5.8 17 22 17 2t 2 Example 5

C
Concentration (in parts per million)
24 22 20 18 16 14 12 10 8 6 4 2 0

10. Medical Dosage. Referring to Example 5, use only the graph showing medical dosage to estimate the value of the polynomial when t 26.
(20, 22)

20.2

(10, 17)

(30, 17)

(2, 5.8) (0, 2)
10 14 20 30 40

ISBN:0-536-47742-6

t

Time (in hours)

Do Exercises 9 and 10.

Answers on page A-16 247
4.3 Introduction to Polynomials

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

CALCULATOR CORNER

Evaluating Polynomials (Note: If you set your graphing calculator in Sci (scientific) mode to do the exercises in Section 4.2, return it to Normal mode now.) There are several ways to evaluate polynomials on a graphing calculator. One method uses a table. To evaluate the polynomial in Example 2(b), x 2 3x 1, when x 4, we first enter y1 x 2 3x 1 on the equationeditor screen. Then we set up a table in ASK mode (see p. 101) and enter the value 4 for x. We see that when x 4, the value of Y1 is 3. This is the value of the polynomial when x 4.
10

X 4

Y1 3
10

Y1

X2

3x + 1

10

X

X

4
10

Y

3

We can also use the Value feature from the CALC menu to evaluate this polynomial. First, we graph y1 x 2 3x 1 in a window that includes the x-value 4. We will use the standard window (see p. 180). Then we press F m 1 or F m [ to access the CALC menu and select item 1, Value. Now we supply the desired x-value by pressing : 4. We then press [ to see X 4, Y 3 at the bottom of the screen. Thus, 2 when x 4, the value of x 3x 1 is 3.

Exercises: Use the Value feature to evaluate the polynomial for the given values of x.
1. x2
2

3x 5x x
2

1, when x 2, when x 8, when x 7, when x

2, x 3, x 3, x 1, x

0.5, and x 1, and x 1.8, and x 2, and x 2.6 3

4

2. 3x 4.

3. 2x 2 5x

3x

3.4

Find an equivalent polynomial using only additions. 11. 9x 3 4x 5

Identifying Terms
As we saw in Section 1.4, subtractions can be rewritten as additions. For any polynomial that has some subtractions, we can find an equivalent polynomial using only additions.
EXAMPLES

Find an equivalent polynomial using only additions. x 5x 2 4x 7 x 4x 5 2x 6 4x 7
ISBN:0-536-47742-6

12.

2y 3

3y 7

7y

9

6.

5x
5

2

7. 4x

2x

6

Do Exercises 11 and 12.

Answers on page A-16 248
CHAPTER 4: Polynomials: Operations

When a polynomial has only additions, the monomials being added are called terms. In Example 6, the terms are 5x 2 and x. In Example 7, the terms are 4x 5, 2x 6, 4x, and 7.

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

EXAMPLE 8

Identify the terms of the polynomial 3x 12 8x 3 5x.

Identify the terms of the polynomial. 13. 3x 2 6x 1 2

4x 7

Terms: 4x 7, 3x, 12, 8x 3, and 5x.

If there are subtractions, you can think of them as additions without rewriting.
EXAMPLE 9

Identify the terms of the polynomial
6

3t

4

5t

4t

2.

14.

4y 5

7y 2

3y

2

Terms: 3t 4,

5t 6,

4t, and 2.

Do Exercises 13 and 14.

Like Terms
When terms have the same variable and the variable is raised to the same power, we say that they are like terms.
EXAMPLES

Identify the like terms in the polynomial. 15. 4x 3 x3 2

Identify the like terms in the polynomials. 4x 2 2x 3 x2
Same variable and exponent Same variable and exponent 16. 4t 4 Constant terms are like terms because 6 and 8 8x 0. 6x
0

10. 4x 3

5x

Like terms: 4x 3 and 2x 3 Like terms: 4x 2 and x 2 11. 6 3a 2 8 a 5a 8 5a Like terms: 6 and Like terms: a and

9t 3

7t 4

10t 3

Do Exercises 15–17.

Coefficients
The coefficient of the term 5x 3 is 5. In the following polynomial, the color numbers are the coefficients, 3, 2, 5, and 4: 3x 5 2x 3 5x 4.

17. 5x 2

3x

10

7x 2

8x

11

EXAMPLE 12

Identify the coefficient of each term in the polynomial 1 2 x 2 x 8.
18. Identify the coefficient of each term in the polynomial 2x 4 7x 3 8.5x 2 10x 4.

3x 4

4x 3

The coefficient of the first term is 3. The coefficient of the second term is The coefficient of the third term is
ISBN:0-536-47742-6
1 2.

4.

The coefficient of the fourth term is 1. The coefficient of the fifth term is
Do Exercise 18.

8.

Answers on page A-16 249
4.3 Introduction to Polynomials

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Collect like terms. 19. 3x 2 5x 2

Collecting Like Terms
We can often simplify polynomials by collecting like terms, or combining like terms. To do this, we use the distributive laws. We factor out the variable expression and add or subtract the coefficients. We try to do this mentally as much as possible.
EXAMPLES

20. 4x 3

2x 3

2

5

Collect like terms.
3

13. 2x
21. 1 5 x 2 3 5 x 4 4x 2 2x 2

3

6x 7

2 6 x3 4x 3 4x 4 2x 2 11

Using a distributive law

14. 5x 2

2x 4

5 2 x2 7x 2 2x 4

4 4

2 x4

7

11

22. 24

4x 3

24

Note that using the distributive laws in this manner allows us to collect like terms by adding or subtracting the coefficients. Often the middle step is omitted and we add or subtract mentally, writing just the answer. In collecting like terms, we may get 0.
EXAMPLE 15

Collect like terms: 3x 5 3x 5 8 3 0x
5

2x 2 2x 2 8
2

3x 5 8

8.

23. 5x 3

8x 5

8x 5

3x 5

2x 2

3 x5 2x 8

2x 2
Do Exercises 19–24.

24.

2x 4

16

2x 4

9

3x 5

Expressing a term like x 2 by showing 1 as a factor, 1 x 2, may make it easier to understand how to factor or collect like terms.
Collect like terms. 25. 7x x EXAMPLES

Collect like terms. 5x 2 1x 2 5 1 x2 6x 2 x8
Replacing x 2 with 1x 2 Using a distributive law

16. 5x 2

x2

17. 5x 8
26. 5x 3 x3 4

6x 5

5x 8 6x 5 5 1 x8 4x 8 6x 5
2 3 5x 3 3 10 x

1x 8 6x 5
2 3 4 6 3 4 6x 1 4 2x 1 6 1 6

x8

1x 8

18. 2x 4 3

x3

1 4 6x

x4 x
4
9 3 10 x 9 3 10 x

1
10 10

2 5 4 10

3 10

x3 x
3

x3

1 x3

3 10

27.

3 3 x 4

4x 2

x3

7 Do Exercises 25–28.

Descending and Ascending Order
4 28. x 4 5 x
4

x

5

1 5

1 4 x 4

10

Note in the following polynomial that the exponents decrease from left to right. We say that the polynomial is arranged in descending order: 2x 4 8x 3 5x 2 x 3.

ISBN:0-536-47742-6

Answers on page A-16 250
CHAPTER 4: Polynomials: Operations

The term with the largest exponent is first. The term with the next largest exponent is second, and so on. The associative and commutative laws allow us to arrange the terms of a polynomial in descending order.

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

EXAMPLES

Arrange the polynomial in descending order. x2 8x 2 2x 3 5x 4x 7 3x 3 6x 5 4x 5 2x 3 3x 3 x2 8x 2 5x
2 3

19. 6x 5 20.
2 3

4x 7 4x 5

Arrange the polynomial in descending order. 29. x 3x 5 4x 3 6x 7 2x 4 5x 2

Do Exercises 29–31. EXAMPLE 21

Collect like terms and then arrange in descending order: 3 3 x2 x
2

2x 2 2x
2

4x 3 4x
3

2x 3. 2x
3

30. 4x 2

3

7x 5

2x 3

5x 4

x

2

6x
3

3

3
2

Collecting like terms

6x

x

3

Arranging in descending order

Do Exercises 32 and 33. 31. 14

7t 2

10t 5

14t 7

We usually arrange polynomials in descending order, but not always. The opposite order is called ascending order. Generally, if an exercise is written in a certain order, we give the answer in that same order.

Degrees
The degree of a term is the exponent of the variable. The degree of the term 5x 3 is 3.
EXAMPLE 22 Collect like terms and then arrange in descending order. 32. 3x 2 2x 3 5x 2 1 x

Identify the degree of each term of 8x 4 3x is 1. x 1.

3x

7.

The degree of 8x 4 is 4. The degree of
Recall that x

The degree of 7 is 0.

Think of 7 as 7x 0. Recall that x 0

1. 33. x

1 2

14x 4

7x

1

4x 4

The degree of a polynomial is the largest of the degrees of the terms, unless it is the polynomial 0. The polynomial 0 is a special case. We agree that it has no degree either as a term or as a polynomial. This is because we can express 0 as 0 0x 5 0x 7, and so on, using any exponent we wish.
EXAMPLE 23

Identify the degree of the polynomial 5x 3 7.
The largest exponent is 4.

6x 4

7.

Identify the degree of each term and the degree of the polynomial. 34. 6x 4 8x 2 2x 9

5x 3

6x 4

The degree of the polynomial is 4.
Do Exercises 34 and 35.

Let’s summarize the terminology that we have learned, using the polynomial 3x 4 8x 3 5x 2 7x 6.
DEGREE OF THE TERM DEGREE OF THE POLYNOMIAL

35. 4

x3

1 6 x 2

x5

ISBN:0-536-47742-6

TERM

COEFFICIENT

3x 4 8x 3 5x 2 7x 6

3 8 5 7 6

4 3 2 1 0

4

Answers on page A-16 251
4.3 Introduction to Polynomials

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Identify the missing terms in the polynomial. 36. 2x
3

Missing Terms
If a coefficient is 0, we generally do not write the term. We say that we have a missing term.
EXAMPLE 24

4x

2

2

Identify the missing terms in the polynomial 5x 2 7x
4

37.

3x 4

8x 5

2x 3

8.

There is no term with x . We say that the x 4-term is missing.
38. x 3 1 Do Exercises 36–39.

For certain skills or manipulations, we can write missing terms with zero coefficients or leave space.
39. x 4 x2 3x 0.25

Write the polynomial x 4 6x 3 its missing terms and by leaving space for them.
EXAMPLE 25

2x 1 1

1 in two ways: with
Writing with the missing x 2-term Leaving space for the missing x 2-term

a) x 4
Write the polynomial in two ways: with its missing terms and by leaving space for them. 40. 2x 3 4x 2 2

6x 3 6x 3

2x 2x

1 1

x4 x4

6x 3 6x 3

0x 2

2x 2x

b) x 4

EXAMPLE 26

Write the polynomial y 5 terms and by leaving space for them. a) y 5 1 1 y5 y
5

1 in two ways: with its missing 1 1

0y 4

0y 3

0y 2

0y

41. a 4

10

b) y

5

Do Exercises 40 and 41.

Classify the polynomial as a monomial, binomial, trinomial, or none of these. 42. 5x 4

Classifying Polynomials
Polynomials with just one term are called monomials. Polynomials with just two terms are called binomials. Those with just three terms are called trinomials. Those with more than three terms are generally not specified with a name.
EXAMPLE 27

43. 4x 3

3x 2

4x

2
MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

44. 3x 2

x

4x 2 9 23x 19

2x 4 3x 5 6x 9x 7 6

3x 3 6x 7 4x 2

4x 7 7x 2 4 6x 1 2

4x 3 5x 2 x 8 z 5 2z 4 z 3 7z 3 4x 6 3x 5 x 4 x 3 2x

1

Do Exercises 42–45. 45. 3x
2

2x

4
ISBN:0-536-47742-6

Answers on page A-16 252
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.3
1. 5x
2

EXERCISE SET

For Extra Help
1. 1 5x 0. 1 x 4 3x 3 7x 2 3x 2
2

MathXL

MyMathLab

InterAct Math

Math Tutor Digital Video Center Tutor CD 2 Videotape 4

Student’s Solutions Manual

Evaluate the polynomial when x 2 x 7

4 and when x 2. 5. x 8x
3

3. 2x 2 x 6. 7 x

5x 3x
2

7

4. 3x

Evaluate the polynomial when x 7. 1 x 3 6 5 x2

2 and when x 8. 8 11.

9. x 2 12. 2x 3

2x

1 5x 2 4x 3

10. 5x

13. Skydiving. During the first 13 sec of a jump, the number of feet S that a skydiver falls in t seconds can be approximated by the polynomial equation S 11.12t 2. Approximately how far has a skydiver fallen 10 sec after having jumped from a plane?

14. Skydiving. For jumps that exceed 13 sec, the polynomial equation S 173t 369 can be used to approximate the distance S , in feet, that a skydiver has fallen in t seconds. Approximately how far has a skydiver fallen 20 sec after having jumped from a plane?

11.12t 2

15. Electricity Consumption. The net consumption of electricity in China can be approximated by the polynomial equation E 90.28t 1138.34, where E is the consumption of electricity, in billions of kilowatt-hours, and t is the number of years since 2001—that is, t 0 corresponds to 2001, t 9 corresponds to 2010, and so on.
Source: Energy Information Administration (EIA), International Energy Outlook 2004

16. Electricity Consumption. The net consumption of electricity in the United States can be approximated by the polynomial equation E 75.72t 3378.11, where E is the consumption of electricity, in billions of kilowatt-hours, and t is the number of years since 2001—that is, t 0 corresponds to 2001, t 9 corresponds to 2010, and so on.
Source: Energy Information Administration (EIA), International Energy Outlook 2004

a) Approximate the consumption of electricity, in billions of kilowatt-hours, in 2001, 2005, 2010, 2015, and 2025. b) Check the results of part (a) using the graph below.
E
Electricity consumption (in billions of kilowatt-hours)
6000 5000 4000 3000 2000 1000 0 4 8 12 16 20 24 E 90.28t 1138.34

a) Approximate the consumption of electricity, in billions of kilowatt-hours, in 2001, 2005, 2010, 2015, and 2025. b) Check the results of part (a) using the graph below.
E
Electricity consumption (in billions of kilowatt-hours)
6000 5000 4000 3000 2000 1000 0 4 8 12 16 20 24 E 75.72t 3378.11

ISBN:0-536-47742-6

t

t

Years since 2001

Years since 2001

253
Exercise Set 4.3
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

17. Total Revenue. Hadley Electronics is marketing a new kind of plasma TV. The firm determines that when it sells x TVs, its total revenue R (the total amount of money taken in) will be R 280x 0.4x 2 dollars. What is the total revenue from the sale of 75 TVs? 100 TVs?

18. Total Cost. Hadley Electronics determines that the total cost C of producing x plasma TVs is given by C 5000 0.6x 2 dollars. What is the total cost of producing 500 TVs? 650 TVs?

19. The graph of the polynomial equation y 5 x 2 is shown below. Use only the graph to estimate the value of the polynomial when x 3, x 1, x 0, x 1.5, and x 2. y 5 4 3 2 1 5 4 3 2 1 1 2 3 4 5

20. The graph of the polynomial equation y 6x 3 6x is shown below. Use only the graph to estimate the value of the polynomial when x 1, x 0.5, x 0.5, x 1, and x 1.1. y 5 4 3 1

y

5

x

2

y

6x 3

6x

1 2 3 4 5

x

5 4 3 2

1 2 3 4 5

2 3 4 5

x

Hearing-Impaired Americans. The number N, in millions, of hearing-impaired Americans of age x can be approximated by the polynomial equation N 0.00006x 3 0.006x 2 0.1x 1.9. The graph of this equation is shown at right. Use either the graph or the polynomial equation for Exercises 21 and 22.
Source: American Speech-Language Hearing Association

N
5 4 3 2 1 0

N

0.00006x 3

0.006x 2

0.1x

1.9

21. Approximate the number of hearing-impaired Americans of ages 20 and 40.

0

20

40

60

80

x

22. Approximate the number of hearing-impaired Americans of ages 50 and 60.
Copyright © 2007 Pearson Education, Inc.

ISBN:0-536-47742-6

254
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Memorizing words. Participants in a psychology experiment were able to memorize an average of M words in t minutes, where M 0.001t 3 0.1t 2. Use the graph below for Exercises 23–28.
M
Number of words memorized
20 18 16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

23. Estimate the number of words memorized after 10 min. 24. Estimate the number of words memorized after 14 min. 25. Find the approximate value of M for t 26. Find the approximate value of M for t 27. Estimate the value of M when t is 13. 28. Estimate the value of M when t is 7. 8. 12.

M

0.001t 3

0.1t 2

t

Time (in minutes)

Identify the terms of the polynomial. 29. 2 3x x2 30. 2x 2 3x 4

31.

2x 4

1 3 x 3

x

3

32.

2 5 x 5

x3

6

Identify the like terms in the polynomial. 33. 5x 3 6x 2 3x 2 34. 3x 2 4x 3 2x 2

35. 2x 4

5x

7x

3x 4

36.

3t

t3

2t

5t 3

37. 3x 5

7x

8

14x 5

2x

9

38. 8x 3

7x 2

11

4x 3

8x 2

29

Identify the coefficient of each term of the polynomial. 39. 3x 6 40. 2x 4 41. 5x 2 3 x 4 3

42.

2 2 x 3

5x

2

43.

5x 4

6x 3

2.7x 2

8x

2

44. 7x 3

4x 2

4.2x

5

Collect like terms. 45. 2x
ISBN:0-536-47742-6

5x

46. 2x 2

8x 2

47. x

9x

48. x

5x

49. 5x 3

6x 3

4

50. 6x 4

2x 4

5

255
Exercise Set 4.3
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

51. 5x 3

6x

4x 3

7x

52. 3a 4

2a

2a

a4

53. 6b 5

3b 2

2b 5

3b 2

54. 2x 2

6x

3x

4x 2

55.

1 5 x 4

5

1 5 x 2

2x

37

56.

1 3 x 3

2x

1 3 x 6

4

16

57. 6x 2

2x 4

2x 2

x4

4x 2

58. 8x 2

2x 3

3x 3

4x 2

4x 2

59.

1 3 x 4

x2

1 2 x 6

3 3 x 8

5 3 x 16

60.

1 4 x 5

1 5

2x 2

1 10

3 4 x 15

2x 2

3 10

Arrange the polynomial in descending order. 61. x 5 x 6x 3 1 2x 2 62. 3 2x 2 5x 6 2x 3 3x

63. 5y 3

15y 9

y

y2

7y 8

64. 9p

5

6p 3

5p 4

p5

Collect like terms and then arrange in descending order. 65. 3x 4 5x 6 2x 4 6x 6 66. 1 5x 3 3 7x 3 x4 5

67.

2x

4x 3

7x

9x 3

8

68.

6x 2

x

5x

7x 2

1

69. 3x

3x

3x

x2

4x 2

70.

2x

2x

2x

x3

5x 3

71.

x

3 4

15x 4

x

1 2

3x 4

72. 2x

5 6

4x 3

x

1 3

2x

73. 2x

4

74. 6

3x

75. 3x 2

5x

2

76. 5x 3

2x 2

3

77.

7x 3

6x 2

3 x 5

7

78. 5x 4

1 2 x 4

Copyright © 2007 Pearson Education, Inc.

Identify the degree of each term of the polynomial and the degree of the polynomial.

ISBN:0-536-47742-6

x

2

79. x 2

3x

x6

9x 4

80. 8x

3x 2

9

8x 3

256
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

81. Complete the following table for the polynomial
TERM COEFFICIENT

7x 4

6x 3

3x 2

8x

2.

DEGREE OF THE TERM

DEGREE OF THE POLYNOMIAL

7x 4 6x 3 6 2 8x 2 1

82. Complete the following table for the polynomial 3x 2
TERM COEFFICIENT

8x 5

46x 3

6x

2.4

1 4 2x .

DEGREE OF THE TERM

DEGREE OF THE POLYNOMIAL

5
1 4 2x

4 46

3x 2 6 2.4

2

Identify the missing terms in the polynomial. 83. x 3 86. 5x 4 27 84. x 5 87. 2x 3 x 5x 2 85. x 4 6x 3 x

7x

2

x

3

88.

Write the polynomial in two ways: with its missing terms and by leaving space for them. 89. x 3 92. 5x 4 27 90. x 5 93. 2x 3 x 5x 2 91. x 4 6x 3 x

7x

2

x

3

94.

Classify the polynomial as a monomial, binomial, trinomial, or none of these. 95. x 2 98. x 2
ISBN:0-536-47742-6

10x

25

96.

6x 4

97. x 3 100. 2x 4

7x 2 7x 3

2x x2

4

9

99. 4x 2

25

x

6

101. 40x

102. 4x 2

12x

9

257
Exercise Set 4.3
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

103.

DW Is it better to evaluate a polynomial before or after like terms have been collected? Why?

104.

DW Explain why an understanding of the rules for order of operations is essential when evaluating polynomials.

SKILL MAINTENANCE
105. Three tired campers stopped for the night. All they had to eat was a bag of apples. During the night, one awoke and ate one-third of the apples. Later, a second camper awoke and ate one-third of the apples that remained. Much later, the third camper awoke and ate one-third of those apples yet remaining after the other two had eaten. When they got up the next morning, 8 apples were left. How many apples did they begin with? [2.6a]

Subtract. [1.4a] 106. 1 20 107. 1 8 5 6 108. 3 8 1 4 109. 5.6 8.2

110. Solve: 3 x

2

5x

9. [2.3c]

111. Solve C

ab

r for b. [2.4b]

112. A nut dealer has 1800 lb of peanuts, 1500 lb of cashews, and 700 lb of almonds. What percent of the total is peanuts? cashews? almonds? [2.5a]

113. Factor: 3x

15y

63. [1.7d]

SYNTHESIS
Collect like terms. 114. 6x 3 7x 2 4x 3
2

3x 3

2

4x 2 5x 3

10x 5

17x 6

115. 3x 2

3

4x 2 4x 4

x 4 2x

2

2x

2 3

100x 2 x 2

2

116. Construct a polynomial in x (meaning that x is the variable) of degree 5 with four terms and coefficients that are integers.

117. What is the degree of 5m 5 2?

Use the CALC feature and choose VALUE on your graphing calculator to find the values in each of the following. 119. Exercise 19 120. Exercise 20 121. Exercise 21 122. Exercise 22

Copyright © 2007 Pearson Education, Inc.

118. A polynomial in x has degree 3. The coefficient of x 2 is 3 less than the coefficient of x 3. The coefficient of x is three times the coefficient of x 2. The remaining coefficient is 2 more than the coefficient of x 3. The sum of the coefficients is 4. Find the polynomial.

ISBN:0-536-47742-6

258
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.4

ADDITION AND SUBTRACTION OF POLYNOMIALS

Objectives
Add polynomials.

Addition of Polynomials
To add two polynomials, we can write a plus sign between them and then collect like terms. Depending on the situation, you may see polynomials written in descending order, ascending order, or neither. Generally, if an exercise is written in a particular order, we write the answer in that same order.
EXAMPLE 1

Simplify the opposite of a polynomial. Subtract polynomials. Use polynomials to represent perimeter and area.

Add: 2x 3 x3 4 4x 3x 2

3x
3

3

2x 4x 3 3x
2

4 3x 2 2x 2

4x 4

3

3x 2

2

2.

3x 3

Collecting like terms

2x

2

Add. 1. 3x 2 2x 2 2x 2 5x 5

EXAMPLE 2
2 4 3x

Add: 3x 2 2x
1 2 1 4 3x

5x 3

3x 2

3x

1 2

.
2. 4x 5 x 3 7x 4 2x 2 4

We have
2 4 3x

3x
2 3

2
1 3

2x x4 5x 3

1 2

1 4 3x

5x 3 x2

3

3x 2

2

3x 3x

5x 3 x.

3

1 2 1 2

1 2

1 4 3x

Collecting like terms 3. 31x 4 7x 4 x 2 2x 1 5x 3 2x 2

We can add polynomials as we do because they represent numbers. After some practice, you will be able to add mentally.
Do Exercises 1–4. EXAMPLE 3

Add: 3x 2 5x 3 x2
3 2

2x
2

2 3x 3x 4

5x 3 2 4

2x 2

3x

4.

4. 17x 3 15x 3

x2 x2

3x 3x

4 2 3

3x

2

2x 5x
3

2

2x

2x x

2 2

You might do this step mentally. Add mentally. Try to write just the answer. 5. 4x 2 5x 3 2x 2 2x 4

5x 3

Then you would write only this.

Do Exercises 5 and 6.

We can also add polynomials by writing like terms in columns.
EXAMPLE 4

3x

6

5x

5

Add: 9x 5 x 5.
2

2x 3

6x 2

3 and 5x 4

7x 2

6 and
6. 3x 3 5x 3 4x 2 2x 2 5x 3x 3 1 2

We arrange the polynomials with the like terms in columns. 9x 5
ISBN:0-536-47742-6

2x 3 5x
4

3x 6 3x 6

5x 5 4x 5

6x 2 7x 2 x2
6

5x 4

2x 3 4x

3 6 5 14
5

We leave spaces for missing terms. Adding

We write the answer as 3x

5x

4

2x 3

14 without the space.

Answers on page A-17 259

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.4 Addition and Subtraction of Polynomials

Add. 7. x4 9x 4 2x
3

Do Exercises 7 and 8. 5x 6x 2 x2
2

2x 7x

6x 3

4 10 2

Opposites of Polynomials
In Section 1.8, we used the property of 1 to show that we can find the opposite of an expression. For example, the opposite of x 2y 5 of can be written as x 2y 5

8.

3x 3 x x3
3

5x x 2x
2

2 and 5 and 4

by changing the sign of every term: x 2y 5 x 2y 5.

This applies to polynomials as well.
OPPOSITES OF POLYNOMIALS

Simplify. 9. 4x 3 6x 3

To find an equivalent polynomial for the opposite, or additive inverse, of a polynomial, change the sign of every term. This is the same as multiplying by 1.

EXAMPLE 5

Simplify: 3x 4

x2 x2 t3 4 t3 7x 4
2

3x 3x

4. 4

x2
EXAMPLE 6

10.

5x 4

3x 2

7x

5

Simplify: 6t 2 t

6t 2 6t 2
5 3 9x

t t

4. 4 x
5 3 9x

t3
EXAMPLE 7

Simplify:
4
5 3 9x

8x 2 7x
4

67 . 8x 2 x 67

7x
11. 14x 10
1 5 2x

8x

x

67

5x 3

x2

3x

Do Exercises 9–11.

Subtraction of Polynomials
Subtract. 12. 7x 3 2x 4 5x 3 4

Recall that we can subtract a real number by adding its opposite, or additive inverse: a b a b . This allows us to subtract polynomials.
EXAMPLE 8

Subtract: x3 2x 2 4 2x 5 x4 4x 3 3x 2 .

9x 5 We have 9x 5
13. 3x 2 4x 2 5x 4 11x 2

x3 9x 5 9x 5

2x 2 x3 x3

4 2x 2 2x 2

2x 5 4 4

x4 2x 5 2x 5

4x 3 x4

3x 2 4x 3 3x 2 3x 2
Adding the opposite Finding the opposite by changing the sign of each term
ISBN:0-536-47742-6

x4

4x 3

7x 5

x4

5x 3

x2

4.

Adding (collecting like terms)

Answers on page A-17
Do Exercises 12 and 13.

260
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

As with similar work in Section 1.8, we combine steps by changing the sign of each term of the polynomial being subtracted and collecting like terms. Try to do this mentally as much as possible.
EXAMPLE 9

Subtract. 14. 6x 4 3x 2 6 2x 4 5x 3 5x 2 7

Subtract: 9x 5 x
3 5

x3
5

2x
3

2x 5 6 6

5x 3

6.

9x

5

2x x
3

2x 2x

5x
5

9x

2x

5x

3

Finding the opposite by changing the sign of each term

11x 5

4x 3

2x

6

Adding (collecting like terms)

Do Exercises 14 and 15.

We can use columns to subtract. We replace coefficients with their opposites, as shown in Example 8.
EXAMPLE 10

15.

Write in columns and subtract: 6 6 3 6 3 6 3 9 9x 2 5x 3.

3 3 x 2 1 3 x 2

1 2 x 2 1 2 x 2

0.3 4 x 3 1.2

5x 2 a) 5x 2 9x 2 5x 2 9x 2 5x 2 9x 2 4x 2

3x 3x 5x 3x 5x 3x 5x 2x

Writing like terms in columns

b)

Changing signs

c)

Adding

Write in columns and subtract. 16. 4x 3 2x 3 2x 2 3x 2 2x 2 3

If you can do so without error, you can arrange the polynomials in columns and write just the answer, remembering to change the signs and add.
EXAMPLE 11

Write in columns and subtract: 2x x x2
2

x3

x2 x 2x 3 3x
3 3

12 2x 3x 5x 12

2x 3

x2

3x .

Leaving space for the missing term

12.

Changing the signs and adding

Do Exercises 16 and 17.

Polynomials and Geometry
EXAMPLE 12

17. 2x 3 x 2 x 5 4x 3

6x 2 2x 2 4x

Find a polynomial for the sum of the areas of these rectangles.

5
ISBN:0-536-47742-6

4

B A x x

x

C 2 x D 5

Recall that the area of a rectangle is the product of the length and the width. The sum of the areas is a sum of products. We find these products and then collect like terms.
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Answers on page A-17 261
4.4 Addition and Subtraction of Polynomials

18. Find a polynomial for the sum of the perimeters and the areas of the rectangles. x 2x x x x qx

Area of A plus Area of B plus Area of C plus Area of D 4x We collect like terms: 4x 5x x2 10 x2 9x 10. 5x x x 2 5

Do Exercise 18. 19. Lawn Area. An 8-ft by 8-ft shed is placed on a lawn x ft on a side. Find a polynomial for the remaining area. EXAMPLE 13 Lawn Area. A water fountain with a 4-ft by 4-ft square base is placed on a square grassy park area that is x ft on a side. To determine the amount of grass seed needed for the lawn, find a polynomial for the grassy area.

4 ft 4 ft x ft

x ft

x ft 8 ft

8 ft

We make a drawing of the situation as shown here. We then reword the problem and write the polynomial as follows. Area of park Area of base of fountain Area left over Area left over

⎧ ⎪ ⎨ ⎪ ⎩ x x Then x
2

x ft

16 ft

2

Area left over.

Do Exercise 19.

Answers on page A-17

CALCULATOR CORNER

Checking Addition and Subtraction of Polynomials A table set in AUTO mode can be used to perform a partial check that polynomials have been added or subtracted correctly. To check Example 3, we enter y1 3x 2 2x 2 5x 3 2x 2 3x 4 3 2 x x 2. If the addition has been done correctly, and y2 5x the values of y1 and y2 will be the same regardless of the table settings used. A graph can also be used to check addition and subtraction. See the Calculator Corner on p. 270 for the procedure.
0 1 2 3 4 X

⎧ ⎪ ⎨ ⎪ ⎩
4 4
X 2 1

Y1 40 7 2 5 44 145 338 2

Y2 40 7 2 5 44 145 338

Exercises: Use a table to determine whether the sum or difference is correct.
1. 2. x 3x 3
3 2 5 4 3

2x 2x x
2

4 3x 4 2x 7

4x 3 3x
2

3x 2
2

2 4x 6 5 7x 6 4 2x

x3 x
2 3

3x 2 x2 2
5

2x x

2 2
ISBN:0-536-47742-6

3. 5x 4. 9x 5. 3x 6.

7x
3 2 2

2x

3x
5

4x 11x
4

2x 2x 4x 5
4

5x 3x
2 2

3

4x 3
2

2x 6x 3

6

2x

1

x

x

5 2x
3

2x

3x

3x

8

262
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.4
Add. 1. 3x 2 4x

EXERCISE SET

For Extra Help

MathXL

MyMathLab

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Math Tutor Digital Video Center Tutor CD 2 Videotape 4

Student’s Solutions Manual

3

2. 6x

1

7x

2

3.

6x

2

x2

1 2x

3

4. x 2

5 3x

4

8x

9

5. x 2

9

x2

9

6. x 3

x2

2x 3

5x 2

7. 3x 2

5x

10

2x 2

8x

40

8. 6x 4

3x 3

1

4x 2

3x

3

9. 1.2x 3

4.5x 2

3.8x

3.4x 3

4.7x 2

23

10. 0.5x 4

0.6x 2

0.7

2.3x 4

1.8x

3.9

11. 1

4x

6x 2

7x 3

5

4x

6x 2

7x 3

12. 3x 4

6x

5x 2

5

6x 2

4x 3

1

7x

13.

1 4 4x

2 3 3x

5 2 8x

7

3 4 4x

3 2 8x

7

14.

1 9 3x 1 9 5x

1 5 5x 1 4 4x

1 2 2x 3 5 5x

7
3 2 4x 1 2

15. 0.02x 5 0.2x 3 x 0.08 0.01x 5 x 4 0.8x 0.02

16. 0.03x 6 7 6 100 x

0.05x 3
3 3 100 x

0.22x 0.5

0.05

17. 9x 8 7x 4 2x 2 5 3x 4 6x 2 2x 1

8x 7

4x 4

2x

18. 4x 5 6x 3 9x 1 4x 3 8x 2 3x 2

6x 3

9x 2

9x

19.

0.15x 4 1.25x 4

0.10x 3 0.01x 3 0.27x 3

0.9x 2 0.01x 2 0.11x 2 15x 2

20. x 0.01 0.99 0.03

0.05x 4 1.5x 4

0.12x 3 0.02x 3 0.25x 3

0.5x 2 0.02x 2 0.01x 2 10x 2

2x 0.15 0.85 0.04

ISBN:0-536-47742-6

0.35x 4

0.25x 4

263
Exercise Set 4.4
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Simplify. 21. 5x 22. x2 3x 23. x2
3 2x

2

24.

4x 3

x2

1 4x

25.

12x 4

3x 3

3

26.

4x 3

6x 2

8x

1

27.

3x

7

28.

2x

4

29.

4x 2

3x

2

30.

6a3

2a2

9a

1

31.

4x 4

6x 2

3 4x

8

32.

5x 4

4x 3

x2

0.9

Subtract. 33. 3x 2 4x 3 34. 6x 1 7x 2 35. 6x 2 x2 x 3

36. x 2

5x

4

8x

9

37. x 2

9

x2

9

38. x 3

x2

2x 3

5x 2

39. 6x 4

3x 3

1

4x 2

3x

3

40.

4x 2

2x

3x 3

5x 2

3

41. 1.2x 3

4.5x 2

3.8x

3.4x 3

4.7x 2

23

42. 0.5x 4

0.6x 2

0.7

2.3x 4

1.8x

3.9

43.

45. 0.08x 3

0.02x 2

0.01x

0.02x 3

0.03x 2

1

46. 0.8x 4

0.2x

1

7 4 10 x

1 5x

0.1

Copyright © 2007 Pearson Education, Inc.

5 3 8x

1 4x

1 3

1 3 8x

1 4x

1 3

44.

1 3 5x

2x 2

0.1

2 3 5x

2x 2

0.01

ISBN:0-536-47742-6

264
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Subtract. 47. x2 x2 5x 2x 6 48. x3 x3 1 x
2

49.

5x 4 6x 4

6x 3 6x 3

9x 2 8x 9

50.

5x 4 6x 3

6x 2 7x 2

3x 8x

6 9

51.

x5 x5

x

4

x

3

x

2

x

1 1

52.

x5 x5

x4 x4

x3 x3

x2 x2

x x

2 2

Solve. Find a polynomial for the perimeter of the figure. 53.
4a

54.
7

3y 3

3a a 2a 5

a
1 2a

7y

2y

7 7 5

2y

3

55. Find a polynomial for the sum of the areas of these rectangles.

56. Find a polynomial for the sum of the areas of these circles.

x 3x x

x x

x

r

3

2

x 4

Find two algebraic expressions for the area of each figure. First, regard the figure as one large rectangle, and then regard the figure as a sum of four smaller rectangles. 57. r 11

58.

t

5

59.

x

3

60.

x

10

9

t

x

8 x

r
ISBN:0-536-47742-6

3

3

265
Exercise Set 4.4
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Find a polynomial for the shaded area of the figure. 61. r 5 m 8 z

62.

m

63.

z 2

64.
7 r 7 7 7

5

16

65.

DW Is the sum of two binomials ever a trinomial? Why or why not?

66.

DW Which, if any, of the commutative, associative, and distributive laws are needed for adding polynomials? Why?

SKILL MAINTENANCE
Solve. [2.3b] 67. 8x 70. 5x 3x 4 66 26 x 68. 5x 71. 1.5x 7x 38 22 5.6x
3 69. 8 x 1 4 3 4x 11 16

x 4

2.7x

72. 3x

3

4x

Solve. [2.3c] 73. 6 y 3 8 4 y 2 5 74. 8 5x 2 7 6x 3

Solve. [2.7e] 75. 3x 7 5x 13 76. 2 x 4 5x 3 7

SYNTHESIS
Find a polynomial for the surface area of the right rectangular solid. 77.
7 x 3 w x 9 a 7

78.

79.
5

x

x

80.
4

81. Find y

Simplify. 82. 3x 2 83. 7y 2 4x 5y x2 7y 3 6 6 2x 3 y2 2x 2 3y 2 6 2y 4 4 8y x 12 3x 3 5y 2 5x 3 8y 2 x2 6y 3 10y 5x 3 y2 3

2

Copyright © 2007 Pearson Education, Inc.

2 2 using the four parts of this square.

ISBN:0-536-47742-6

y

84. 85.

4 y4

2 y

266
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.5

MULTIPLICATION OF POLYNOMIALS

Objectives
Multiply monomials.

We now multiply polynomials using techniques based, for the most part, on the distributive laws, but also on the associative and commutative laws. As we proceed in this chapter, we will develop special ways to find certain products.

Multiply a monomial and any polynomial. Multiply two binomials. Multiply any two polynomials.

Multiplying Monomials
Consider 3x 4x . We multiply as follows: 3x 4x 3 x 4 x 3 4 x x 3 4 x x 12x 2.
MULTIPLYING MONOMIALS

By the associative law of multiplication By the commutative law of multiplication By the associative law Using the product rule for exponents Multiply. 1. 3x 5

To find an equivalent expression for the product of two monomials, multiply the coefficients and then multiply the variables using the product rule for exponents.

2.

x

x

EXAMPLES

Multiply. 5 6 x x 30x 2
By the associative and commutative laws Multiplying the coefficients and multiplying the variables

3.

x

x

1. 5x 6x

4.

x2 x3

2. 3x 3.

x

3x 3

1x 1 x x 7 4 x5 x3 28x 5 28x 8
3

3x 2
5. 3x 5 4x 2

7x 5 4x 3

Adding the exponents Simplifying

After some practice, you can do this mentally. Multiply the coefficients and then the variables by keeping the base and adding the exponents. Write only the answer.
Do Exercises 1–8.

6. 4y 5

2y 6

7.

7y 4

y

Multiplying a Monomial and Any Polynomial
To find an equivalent expression for the product of a monomial, such as 2x, and a binomial, such as 5x 3, we use a distributive law and multiply each term of 5x 3 by 2x.
EXAMPLE 4 8. 7x 5 0

ISBN:0-536-47742-6

Multiply: 2x 5x 3 2x 5x 10x 2 6x

3. 2x 3

2x 5x

Using a distributive law Multiplying the monomials

Answers on page A-17 267
4.5 Multiplication of Polynomials

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Multiply. 9. 4x 2x 4

EXAMPLE 5

Multiply: 5x 2x 2

3x

4.

5x 2x 2
10. 3t
2

3x

4

5x 2x 2 10x
3

5x 3x
2

5x 4

15x

20x

5t

2
MULTIPLYING A MONOMIAL AND A POLYNOMIAL

11.

5x 3 x 3

5x 2

6x

8

To multiply a monomial and a polynomial, multiply each term of the polynomial by the monomial.

12. Multiply:

y

2

y

7.

EXAMPLE 6

Multiply:
3

2x 2 x 3 10x 4

7x 2

10x

4. 2x 2 4

a) Fill in the blanks in the steps of the solution below. y 2 y y 2 y 7 2 y 2

2x x

2

7x 2x
2

2

x

3

2x 2 7x 2 20x 3 8x 2

2x 2 10x

2x 5
Do Exercises 9–11.

14x 4

Multiplying Two Binomials y2 14

To find an equivalent expression for the product of two binomials, we use the distributive laws more than once. In Example 7, we use a distributive law three times.
EXAMPLE 7

Multiply: x

5 x

4.

b) Write an algebraic expression that represents the total area of the four smaller rectangles in the figure shown here.
2

x

5 x

⎫ ⎬ ⎭
4 xx x x x2 x2

4

5x x 4

4 5 x 20 5 4

Using a distributive law Using a distributive law on each part

4x 9x

5x 20

Multiplying the monomials Collecting like terms

y

y

7

To visualize the product in Example 7, consider a rectangle of length x and width x 4.
4 4x 20

5

Multiply. 13. x 8 x 5

x

4 x

x2

5x

x x 5

5

14. x

5 x

4

ISBN:0-536-47742-6

The total area can be expressed as x smaller areas, x 2 4x 5x 20, or x 2

5 x 4 or, by adding the four 9x 20.

Answers on page A-17 268
CHAPTER 4: Polynomials: Operations

Do Exercises 12–14.

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

EXAMPLE 8

Multiply: 4x

3 x

2.

Multiply. 15. 5x 3 x 4

4x

3 x

⎫ ⎬ ⎭
2 4x x 4x x 4x 2 4x
2

2

3x 4x 2

2 3 x 6 3 2

Using a distributive law Using a distributive law on each part

8x 5x

3x 6

Multiplying the monomials Collecting like terms

Do Exercises 15 and 16.

Multiplying Any Two Polynomials
Let’s consider the product of a binomial and a trinomial. We use a distributive law four times. You may see ways to skip some steps and do the work mentally.
EXAMPLE 9 16. 2x 3 3x 5

Multiply: x 2

2x

3 x2

4.

x2

2x

3 x2

⎫ ⎬ ⎭ x2 x2 x x x
Do Exercises 17 and 18.
PRODUCT OF TWO POLYNOMIALS
2 4 4

4

4
2 2 3

2x x 2 x
2

4
2

3 x2 2x 4
2

4 3 x2 3 4

x

4 2x x
2 3

2x x 8x 8x

4x 2x

3x 12

12

Multiply. 17. x 2 3x 4 x2 5

To multiply two polynomials P and Q, select one of the polynomials— say, P. Then multiply each term of P by every term of Q and collect like terms.

To use columns for long multiplication, multiply each term in the top row by every term in the bottom row. We write like terms in columns, and then add the results. Such multiplication is like multiplying with whole numbers. 321 12 642 321 3852
EXAMPLE 10

300 600 200 800

20 10 40 10 50 2x 2

1 2 2 2
Multiplying the top row by 2 Multiplying the top row by 10 Adding

18. 3y 2

7 2y 3

2y

5

3000 3000

Multiply: 4x 3
3

3x x 2

2x .

4x 8x 4 2x 4 6x 4

2x x2 4x 3 3x 3 x3

2

3x 2x 6x 2 6x 2
Multiplying the top row by 2x Multiplying the top row by x 2 Collecting like terms Line up like terms in columns.

ISBN:0-536-47742-6

4x

5

4x 5

Answers on page A-17 269
4.5 Multiplication of Polynomials

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Multiply. 19. 3x
2

EXAMPLE 11 2x x 4 5

Multiply: 5x 3

3x

4

2x 2

3.

When missing terms occur, it helps to leave spaces for them and align like terms as we multiply. 5x 3 2x 2 10x
5

3x 9x 8x
2

4 3 12 12
Multiplying by Multiplying by 3 2x 2

20.

5x 2

4x 4x 2

2 8

15x 3 6x 3 9x 3

10x 5

8x 2

9x

Collecting like terms

Do Exercises 19 and 20. EXAMPLE 12 21. Multiply. 3x 2 2x
2

Multiply: 2x 2 2x 2x 2
2

3x 4 3 12

4 2x 2

x

3.

2x x

5 2

3x x 9x 4x 13x

4x 4 4x 4

2x 6x 3 4x 3

3

6x 2 3x 2 8x 2 5x 2

Multiplying by 3 Multiplying by x Multiplying by 2x 2

12

Collecting like terms

Answers on page A-17 CALCULATOR CORNER

Do Exercise 21.

Checking Multiplication of Polynomials A partial check of multiplication of polynomials can be performed graphically on the TI-84 Plus graphing calculator. Consider the product x 3 x 2 x 2 x 6. We will use two graph styles to determine whether this product is correct. First, we press G to determine whether SEQUENTIAL mode is selected. If it is not, we position the blinking cursor over SEQUENTIAL and then press [. Next, on the Y screen, we enter y1 x 3 x 2 and y2 x 2 x 6. We will select the line-graph style for y1 and the path style for y2. To select these graph styles, we use f to position the cursor over the icon to the left of the equation and press [ repeatedly until the desired style of icon appears, as shown below. y1 NORMAL SCI ENG FLOAT 0123456789 RADIAN DEGREE FUNC PAR POL SEQ CONNECTED DOT SEQUENTIAL SIMUL REAL a bi re^ i FULL HORIZ G T
Plot1 Plot2 Plot3

(x

3)(x

2), y2
10

x2

x

6

\Y1 Y2 \Y3 \Y4 \Y5 \Y6 \Y7

(X 3)(X 2) X2 X 6
10 10

10

The graphing calculator will graph y1 first as a solid line. Then it will graph y2 as the circular cursor traces the leading edge of the graph, allowing us to determine visually whether the graphs coincide. In this case, the graphs appear to coincide, so the factorization is probably correct. A table can also be used to perform a partial check of a product. See the Calculator Corner on p. 262 for the procedure.

ISBN:0-536-47742-6

Exercises: Determine graphically whether the product is correct.
1. x 2. 4x 5 x 3 x 4 2 x2 4x
2

9x 5x

20 6

3. 5x 4. 2x

3 x 3 3x

4 5

5x 2 6x 2

17x 19x

12 15

270
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.5
Multiply. 1. 8x 2 5 5. 8x 5 4x 3
1 3 5x 1 3x

EXERCISE SET

For Extra Help

MathXL

MyMathLab

InterAct Math

Math Tutor Digital Video Center Tutor CD 2 Videotape 4

Student’s Solutions Manual

2. 4x 2

2

3.

x2

x

4.

x3 x2 0.8x 6

6. 10a 2 2a 2
1 4 4x 1 8 5x

7. 0.1x 6 0.3x 5 4x 2 0

8. 0.3x 4 4m 5

9.

10.

11.

12.

1

13. 3x 2

4x 3 2x 6

14.

2y 5 10y 4

3y 3

Multiply. 15. 2x x 5 1 16. 3x 4x 20. 2x 3 x 2 4x 2 x 2 6 1 17. 5x x 1 6x 1 8y 3 18. 22. 3x x 1 6x 2 6y 2 5x 1

19. x 2 x 3

21. 3x 2x 2 25. 3y 2 6y 4

4x 2x 3 y3

23.

6x 2 x 2

x

24.

x

26. 4y 4

Multiply. 27. x 31. x 35. 5 6 x 4 x x 5
5 2

3 3 2x
2 5

28. x 32. x 36. 3

5 x 7 x x 6
4 3

2 3 2x
3 2

29. x 33. x 37. 2x

5 x 3 x 5 2x

2 3 5

30. x 34. x 38. 3x

6 x 6 x 4 3x

2 6 4

39. x

x

40. x

x

41. x

2.3 x

4.7

42. 2x

0.13 2x

0.13

Write an algebraic expression that represents the total area of the four smaller rectangles. 43.
2 x

44.
1 x

45.
1 x

46.
3

x x 6 x 7 x 6 x 8

Draw and label rectangles similar to the one following Example 7 to illustrate each product.
ISBN:0-536-47742-6

47. x x 50. x

5 3 x 1

48. x x 51. x

2 5 x 3

49. x 52. x

1 x 4 x

2 6

271
Exercise Set 4.5
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Multiply. 53. x 2 x 1 x 1 54. x 2 x 2 x 2 55. 2x 1 2x 2 6x 1

56. 3x

1 4x 2

2x

1

57. y 2

3 3y 2

6y

2

58. 3y 2

3

y2

6y

1

59. x 3

x2 x3

x2

x

60. x 3

x2 x3

x2

x

61.

5x 3

7x 2

1 2x 2

x

62.

4x 3

5x 2

2 5x 2

1

63. 1

x

x2

1

x

x2

64. 1

x

x2 1

x

x2

65. 2t 2

t

4 3t 2

2t

1

66. 3a 2

5a

2 2a 2

3a

4

67. x

x3

x5 x2

1

x4

68. x

x3

x 5 3x 2

3x 6

3x 4

69. x 3

x2

x

1 x

1

70. x

2 x3

x2

x

2

71. x

1 x3

7x 2

5x

4

72. x

2 x3

5x 2

9x

3

73. x

1 2

2x 3

4x 2

3x

2 5

74. x

1 3

6x 3

12x 2

5x

1 2

75.

DW Under what conditions will the product of two binomials be a trinomial?

76.

DW How can the following figure be used to show that x 3
2

x2

9?

SKILL MAINTENANCE
Simplify. 77. 1 4 1 2 [1.4a] 78. 3.8 [1.4a] 10.2 79. 10 2 10 [1.8d] 2 80. 10 2 [1.8d] 6
2
Copyright © 2007 Pearson Education, Inc.

3 2

ISBN:0-536-47742-6

Factor. [1.7d] 81. 15x 18y 12 1 x 2 3. [3.2a] 82. 16x 24y 36 83. 9x 45y 15 3 52 3x 84. 100x 1. [2.3c] 100y 1000a

85. Graph: y

86. Solve: 4 x

272
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

SYNTHESIS
Find a polynomial for the shaded area of the figure. 87.
14y 5 3y 6y 3y 5 4t 2t

88.

21t 3t

8 4

89. A box with a square bottom is to be made from a 12-in.square piece of cardboard. Squares with side x are cut out of the corners and the sides are folded up. Find the polynomials for the volume and the outside surface area of the box.

x x

x x

12

x x 12 x

x

For each figure, determine what the missing number must be in order for the figure to have the given area. 90. Area x2 7x
2 ?

10

91. Area

x2

8x

15

x x x ? x 3

92. An open wooden box is a cube with side x cm. The box, including its bottom, is made of wood that is 1 cm thick. Find a polynomial for the interior volume of the cube.
1 cm

93. Find a polynomial for the volume of the solid shown below.
(x 2) m xm

xm x cm 5m

7m

6m x cm x cm

Compute and simplify. 94. x 96. x
ISBN:0-536-47742-6

3 x 5
2

6 x 3

x
2

3 x

6

95. x

2 x

7

x

7 x

2

97. Extend the pattern and simplify: x a x b x c x d x z.

98.

Use a graphing calculator to check your answers to Exercises 15, 29, and 53. Use graphs, tables, or both, as directed by your instructor.

273
Exercise Set 4.5
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Objectives
Multiply two binomials mentally using the FOIL method. Multiply the sum and the difference of two terms mentally. Square a binomial mentally. Find special products when polynomial products are mixed together.

4.6

SPECIAL PRODUCTS

We encounter certain products so often that it is helpful to have faster methods of computing. Such techniques are called special products. We now consider special ways of multiplying any two binomials.

Products of Two Binomials Using FOIL
To multiply two binomials, we can select one binomial and multiply each term of that binomial by every term of the other. Then we collect like terms. Consider the product x 3 x 7 :

x

3 x

⎫ ⎬ ⎭
7 xx x x x2 x2

7 7x 10x

3x x 7 3x 21.

7 3 x 21 3 7

This example illustrates a special technique for finding the product of two binomials:

Study Tips
CHECKLIST
The foundation of all your study skills is TIME! Have you tried using the audio recordings? Are you taking the time to include all the steps when working your homework and the tests? Are you using the timemanagement suggestions we have given so you have the proper amount of time to study mathematics? Have you been using the supplements for the text such as the Student’s Solutions Manual and the AW Math Tutor Center? Have you memorized the rules for special products of polynomials and for manipulating expressions with exponents?

First Outside Inside terms terms terms

Last terms

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

⎫ ⎬ ⎭

x

3 x

7

x x

7 x

3 x

3

7.

To remember this method of multiplying, we use the initials FOIL.
THE FOIL METHOD

To multiply two binomials, A B and C D, multiply the First terms AC, the Outside terms AD, the Inside terms BC, and then the Last terms BD. Then collect like terms, if possible. A 1. 2. 3. 4. B C D AC AD BC F A B C I O D BD L

Multiply First terms: AC. Multiply Outside terms: AD. Multiply Inside terms: BC. Multiply Last terms: BD.

FOIL

EXAMPLE 1

Multiply: x F

8 x2 O x 8x
2

5. I 5
2

We have L 8 5
ISBN:0-536-47742-6

x

8 x2

5

x x2 x x
3 3

8 x2 40 40.

5x 8x

5x

Since each of the original binomials is in descending order, we write the product in descending order, as is customary, but this is not a “must.”

274
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Often we can collect like terms after we have multiplied.
EXAMPLES

Multiply. 6 4 2
3

Multiply mentally, if possible. If you need extra steps, be sure to use them. 1. x 3 x 4

2. x 3. x 4. y 5. x
3

6 x 7 x 3 y 5 x

x2 x2 x2 x2 y2 y 5 2
2

6x 36 4x 11x 2y 5y
6

6x 7x 28 3y 6
3

36 28

Using FOIL Collecting like terms

2. x

3 x

5

6 5x 3 25 15t 2 10
3. 2x 1 x 4

x

5x 25

x6 6. 4t 3 5 3t 2

12t 5

8t 3

4. 2x 2

3 x

2

Do Exercises 1–8. EXAMPLES

Multiply.
2 3

7. x 8. x 2 9. 3

2 3

x

x

2

x2 0.3 x 2 4x 7 0.3 5x 3

2 3x 4 9

2 3x

4 9

5. 6x 2

5 2x 3

1

x4 x 21 21
4

0.3x 2 0.6x 15x 3 28x
2

0.3x 2 0.09 28x 20x 4 20x 4
3

0.09

6. y 3

7 y3

7

15x

7. t

5 t

3

(Note: If the original polynomials are in ascending order, it is natural to write the product in ascending order, but this is not a “must.”) 10. 5x 4 2x 3 3x 2 7x 15x 6 15x
Do Exercises 9–12.
6

35x 5 29x
5

6x 5 14x
4

14x 4

8. 2x 4

x2

x3

x

We can show the FOIL method geometrically as follows. The area of the large rectangle is A B C D. The area of rectangle 1 is AC. The area of rectangle 2 is AD. The area of rectangle 3 is BC. The area of rectangle 4 is BD. The area of the large rectangle is the sum of the areas of the smaller rectangles. Thus, A
ISBN:0-536-47742-6

Multiply. 9. x 4 5 x 4 5

C

D

C D

1 2

3 4

A A

B B

10. x 3

0.5 x 2

0.5

11. 2

3x 2 4

5x 2

B C

D

AC

AD

BC

BD.
12. 6x 3 3x 2 5x 2 2x

Multiplying Sums and Differences of Two Terms
Consider the product of the sum and the difference of the same two terms, such as x 2 x 2.

Answers on page A-18 275
4.6 Special Products

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Multiply. 13. x 5 x 5

Since this is the product of two binomials, we can use FOIL. This type of product occurs so often, however, that it would be valuable if we could use an even faster method. To find a faster way to compute such a product, look for a pattern in the following: a) x b) 3x 2 x 5 3x 2 5 x2 x
2

2x 4; 9x 2 9x
2

2x 15x 25.

4 15x

Using FOIL

25

14. 2x

3 2x

3 Do Exercises 13 and 14.

Perhaps you discovered in each case that when you multiply the two binomials, two terms are opposites, or additive inverses, which add to 0 and “drop out.”
Multiply. 15. x 2 x 2
PRODUCT OF THE SUM AND THE DIFFERENCE

The product of the sum and the difference of the same two terms is the square of the first term minus the square of the second term: A
16. x 7 x 7

B A

B

A2

B 2.

It is helpful to memorize this rule in both words and symbols. (If you do forget it, you can, of course, use FOIL.)
EXAMPLES

Multiply. (Carry out the rule and say the words as you go.) B) 4 A2 x2 x2 B2 42 16
2

(A
17. 6 4y 6 4y

B) (A 4 x

11. x

“The square of the first term, x 2, minus the square of the second, 42” Simplifying
2

12. 5 13. 3x 2
18. 2x
3

2w 5 7 3x 2 10 3 8

2w 7 4x 3 8

5

2w 4w 2
2

25
1 2x
3

3x 2 9x
4

72 49 4x
2

1

14.

4x

10

10 2 100 9 64

16x 2 15.
2 5 2 5

x

x

x2

3 8

2

x2

19.

x

x

Do Exercises 15–19.

Squaring Binomials
ISBN:0-536-47742-6

Answers on page A-18 276
CHAPTER 4: Polynomials: Operations

Consider the square of a binomial, such as x 3 2. This can be expressed as x 3 x 3 . Since this is the product of two binomials, we can again use FOIL. But again, this type of product occurs so often that we would like to use an even faster method. Look for a pattern in the following:

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

a) x

3

2

x x2 x2

3 x 3x 6x 15p 30p

3 3x 9; 3p 15p 9p ;
2

b) x 9 d) 3x 9p
2

3

2

x x2 x2

3 x 3x 6x
2 2

3 3x 9; 5 15x 25. 25 9

Multiply. 20. x 8 x 8

c) 5

3p

2

5 25 25

3p 5

5

2

3x 9x 9x

5 3x 15x 30x

21. x Do Exercises 20 and 21.

5 x

5

When squaring a binomial, we multiply a binomial by itself. Perhaps you noticed that two terms are the same and when added give twice the product of the terms in the binomial. The other two terms are squares.
SQUARE OF A BINOMIAL

Multiply. 22. x 2
2

The square of a sum or a difference of two terms is the square of the first term, plus or minus twice the product of the two terms, plus the square of the last term: A B
2

A2

2AB

B 2;

A

B

2

A2

2AB

B 2.
23. a 4
2

It is helpful to memorize this rule in both words and symbols.
EXAMPLES

Multiply. (Carry out the rule and say the words as you go.) A2 x2 x2 2 A B 2 x 6x 9 B2 52 72
2

(A 16. x

B)2 3
2

B2 32
“x 2 plus 2 times x times 3 plus 32” 24. 2x 5
2

3

(A 17. t 18. 2x 19. 5x 20. 2.3

B)2 5
2

A2 t2 t2

2 A B 2 10t
2

t

5 25

“t 2 minus 2 times t times 5 plus 52” 25. 4x 2 3x
2

7

2 2 2

2x

2 2x 7
2

4x 2 3x
2 2

28x 25x
2

49
2

3x

5x
2

2 5x 3x 24.84m

30x 3

9x 4

5.4m

2.32 5.29

2 2.3 5.4m

5.4m

29.16m 2

26. 7.8

1.2y 7.8

1.2y

Do Exercises 22–27.

Caution!
Although the square of a product is the product of the squares, the square of a sum is not the sum of the squares. That is, AB 2 A2B 2, but The term 2AB is missing.
ISBN:0-536-47742-6

27. 3x 2

5 3x 2

5

A

B

2

A2

B 2.

To confirm this inequality, note, using the rules for order of operations, that 7 whereas 72 52 49 25 74, and 74 144. 5
2

122

144,

Answers on page A-18 277
4.6 Special Products

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

28. In the figure at right, describe in terms of area the sum A2 B 2. How can the figure be used to verify that A B 2 A2 B 2?

We can look at the rule for finding A area of the large square is A B A B A B 2.

B 2 geometrically as follows. The
A B

This is equal to the sum of the areas of the smaller rectangles: A2 Thus, A AB B
2

A

B

A

A2

AB A

AB A2

B2 2AB

A2 B 2.

2AB

B 2.
B AB A A B B2 B B

Do Exercise 28.

Multiplication of Various Types
Let’s now try several types of multiplications mixed together so that we can learn to sort them out. When you multiply, first see what kind of multiplication you have. Then use the best method.
MULTIPLYING TWO POLYNOMIALS

1. Is it the product of a monomial and a polynomial? If so, multiply each term of the polynomial by the monomial. Example: 5x x 7 5x x 5x 7 5x 2 35x 2. Is it the product of the sum and the difference of the same two terms? If so, use the following: A B A B A2 B 2. The product of the sum and the difference of the same two terms is the difference of the squares. [The answer has 2 terms.] Example: x A or A B A B A 7 x B B 7 A A x2 B B
2 2

72 A2 A
2

x2 2AB 2AB

49 B 2, B 2.

3. Is the product the square of a binomial? If so, use the following:

The square of a binomial is the square of the first term, plus or minus twice the product of the two terms, plus the square of the last term. [The answer has 3 terms.]

Answer on page A-18

Example: x

7 x

7

x x2

72 2 x 7

72

x2

14x

49

4. Is it the product of two binomials other than those above? If so, use FOIL. [The answer will have 3 or 4 terms.]

Study Tips
MEMORIZING FORMULAS
Memorizing can be a very helpful tool in the study of mathematics. Don’t underestimate its power as you consider the special products. Consider putting the rules, in words and in math symbols, on index cards and go over them many times.

Example: x

7 x

4

x2

4x

7x

28

x2

3x

28

5. Is it the product of two polynomials other than those above? If so, multiply each term of one by every term of the other. Use columns if you wish. [The answer will have 2 or more terms, usually more than 2 terms.] Example: x2 3x 2 x 7 x2 x 7 x2 x x2 2 x 2 x 3 7x 2 x 3 4x 2 3x x 7 7 3x x 7 3x 2 21x 19x 14 2x 7 3x 7 2x 14
ISBN:0-536-47742-6

278
CHAPTER 4: Polynomials: Operations

Remember that FOIL will always work for two binomials. You can use it instead of either of rules 2 and 3, but those rules will make your work go faster.
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

EXAMPLE 21

Multiply: x 3 x2 9

3 x

3.

Multiply. 29. x 5 x 6

x

3 x

Using method 2 (the product of the sum and the difference of two terms)

EXAMPLE 22

Multiply: t 5 t
2

7 t 2t 35

5.
Using method 4 (the product of two binomials, but neither the square of a binomial nor the product of the sum and the difference of two terms) 30. t 4 t 4

t

7 t

EXAMPLE 23

Multiply: x 6 x2 x2

6 x 26x 12x

6. 36 36 x 7. 14x 3
Using method 1 (the product of a monomial and a trinomial; multiplying each term of the trinomial by the monomial) 32. 9x 2 1
2

x

6 x

Using method 3 (the square of a binomial sum)

31. 4x 2

2x 3

5x 2

10

EXAMPLE 24

Multiply: 2x 3 9x 2 x 7 18x 5

2x 3 9x 2

2x 4

EXAMPLE 25

Multiply: 5x 3
2

7x 2. 49x 2
Using method 3 (the square of a binomial difference)

33. 2a

5 2a

8

5x 3

7x

25x 6

2 5x 3 7x

25x 6
EXAMPLE 26

70x 4
1 2 4 . 1 4

49x 2
1 2
2

Multiply: 3x 9x 2 2 3x

34. Using method 3 (the square of a binomial sum. To get the middle term, we multiply 3x by 1 and 4 double.)

5x

3x

1 2 4

1 16

9x 2
EXAMPLE 27

3 2x

1 16 3 2 4 . 3 4 9 16 9 16

Multiply: 4x 16x 2 16x 2 2 4x 6x

35. Using method 3 (the square of a binomial difference)

2x

1 2

2

4x

3 2 4

EXAMPLE 28

Multiply: p
2

3 p2

2p

1.

36. x 2

x

4 x

2

p
ISBN:0-536-47742-6

2p p 6p p 5p

1 3 3 3

Using method 5 (the product of two polynomials) Multiplying by 3 Multiplying by p

p

3

3p 2 2p 2 5p 2

p3

Answers on page A-18
Do Exercises 29–36.

279
4.6 Special Products

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

1 x x 5

Visualizing for Success
In each of Exercises 1–10, find two algebraic expressions for the shaded area of the figure from the list below.

6
5x

5x

3

3

3

A. 9

4x 2 x 62 3 x 3 22 8x 15 5 x 3 6x 9 2x 2 32 32 2x
2

2
3 x

3

x

B. x 2 C. x D. 102 E. x 2 F. x G. x 2 H. 3 I. x J. 5x K. L. M. N. O. P. Q. 5
2

7 x x 3 3

x x 3 3

4x 3

2x

3 x x

5

4x 5

2x

x 9 104 x2 15

8 x x

5 x x

4

3

12x 36 25x 2 30x 9 x 5 x 3 3x 5 5x R. x 3 2 S. 25 4x 2 T. x 2 6x 9 x 5 3

3

9 x x

x x 3

Answers on page A-18
3 x x x 3 x x

5 x x 2 2

10
3
ISBN:0-536-47742-6

10

x

10

x

3

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.6
1. x 1 x2 3

EXERCISE SET

For Extra Help

MathXL

MyMathLab

InterAct Math

Math Tutor Digital Video Center Tutor CD 2 Videotape 4

Student’s Solutions Manual

Multiply. Try to write only the answer. If you need more steps, be sure to use them. 2. x 2 3 x 1 3. x 3 2 x 1 4. x 4 2 x 10

5.

y

2 y

3

6. a

2 a

3

7. 3x

2 3x

2

8. 4x

1 4x

1

9. 5x

6 x

2

10. x

8 x

8

11. 3t

1 3t

1

12. 2m

3 2m

3

13. 4x

2 x

1

14. 2x

1 3x

1

15. p

1 4

p

1 4

16. q

3 4

q

3 4

17. x

0.1 x

0.1

18. x

0.3 x

0.4

19. 2x 2

6 x

1

20. 2x 2

3 2x

1

21.

2x

1 x

6

22. 3x

4 2x

4

23. a

7 a

7

24. 2y

5 2y

5

25. 1

2x 1

3x

26.

3x

2 x

1

27.

3 8y

5 6

3 8y

5 6

28.

1 5x

2 7

1 5x

2 7

29. x 2
ISBN:0-536-47742-6

3 x3

1

30. x 4

3 2x

1

31. 3x 2

2 x4

2

281
Exercise Set 4.6
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

32. x 10

3 x 10

3

33. 2.8x

1.5 4.7x

9.3

34. x

3 8

x

4 7

35. 3x 5

2 2x 2

6

36. 1

2x 1

3x 2

37. 8x 3

1 x3

8

38. 4

2x 5

2x 2

39. 4x 2

3 x

3

40. 7x

2 2x

7

41. 4y 4

y2 y2

y

42. 5y 6

3y 3 2y 6

2y 3

Multiply mentally, if possible. If you need extra steps, be sure to use them. 43. x 4 x 4 44. x 1 x 1 45. 2x 1 2x 1

46. x 2

1 x2

1

47. 5m

2 5m

2

48. 3x 4

2 3x 4

2

49. 2x 2

3 2x 2

3

50. 6x 5

5 6x 5

5

51. 3x 4

4 3x 4

4

52. t 2

0.2 t 2

0.2

53. x 6

x2 x6

x2

54. 2x 3

0.3 2x 3

0.3
Copyright © 2007 Pearson Education, Inc.

55. x 4

3x x 4

3x

56.

3 4

2x 3

3 4

2x 3

57. x 12

3 x 12

3

ISBN:0-536-47742-6

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CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

58. 12

3x 2 12

3x 2

59. 2y 8

3 2y 8

3

60. m

2 3

m

2 3

61.

5 8x

4.3

5 8x

4.3

62. 10.7

x 3 10.7

x3

Multiply mentally, if possible. If you need extra steps, be sure to use them. 63. x 2
2

64. 2x

1

2

65. 3x 2

1

2

66. 3x

3 2 4

67. a

1 2 2

68. 2a

1 2 5

69. 3

x

2

70. x 3

1

2

71. x 2

1

2

72. 8x

x2

2

73. 2

3x 4

2

74. 6x 3

2

2

75. 5

6t 2

2

76. 3p 2

p

2

77. x

5 2 8

78. 0.3y

2.4

2

Multiply mentally, if possible. 79. 3 2x 3
2

80. x

4x 3

2

81. 4x x 2

6x

3

82. 8x

x5

6x 2

9

83. 2x 2

1 2

2x 2

1 2

84.

x2

1

2

85.

1

3p 1

3p

86.

3q

2 3q

2

87. 3t 2 5t 3
ISBN:0-536-47742-6

t2

t

88.

6x 2 x 3

8x

9

89. 6x 4

4

2

90. 8a

5

2

283
Exercise Set 4.6
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

91. 3x

2 4x 2

5

92. 2x 2

7 3x 2

9

93. 8

6x 4

2

94.

1 2 5x

9

3 2 5x

7

95. t

1 t2

t

1

96. y

5 y2

5y

25

Compute each of the following and compare. 97. 32 42; 3 4
2

98. 62

72; 6

7

2

99. 9 2

52; 9

5

2

100. 112

42; 11

4

2

Find the total area of all the shaded rectangles. 101. 1 102.
3

103. t 4

t

6

104. x 3

x

7

a

x

a

1

x

3

105.

DW Under what conditions is the product of two binomials a binomial?

106.

DW Brittney feels that since the FOIL method can be

used to find the product of any two binomials, she needn’t study the other special products. What advice would you give her?

SKILL MAINTENANCE
107. Electricity Usage. In apartment 3B, lamps, an air conditioner, and a television set are all operating at the same time. The lamps use 10 times as many watts of electricity as the television set, and the air conditioner uses 40 times as many watts as the television set. The total wattage used in the apartment is 2550. How many watts are used by each appliance? [2.6a] Solve. [2.3c] 108. 3x 8x 47 8x 109. 3 x 2 5 2x 7 110. 5 2x 3 2 3x 4 20
Copyright © 2007 Pearson Education, Inc.

ISBN:0-536-47742-6

Solve. [2.4b] 111. 3x 2y 12, for y 112. 3a 5d 4, for a

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CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

SYNTHESIS
Multiply. 113. 5x 3x 1 2x 3 114. 2x 3 2x 3 4x 2 9 115. a 5 a 5
2

116. a 3 2 a 3 2 (Hint: Examine Exercise 115.)

117. 3t 4 2 2 3t 4 2 2 (Hint: Examine Exercise 115.)

118. 3a

2a

3

3a

2a

3

Solve. 119. x 2 x 5 x 1 x 3 120. 2x 5 x 4 x 5 2x 4

121. Factors and Sums. To factor a number is to express it as a product. Since 12 4 3, we say that 12 is factored and that 4 and 3 are factors of 12. In the following table, the top number has been factored in such a way that the sum of the factors is the bottom number. For example, in the first column, 40 has been factored as 5 8, and 5 8 13, the bottom number. Such thinking is important in algebra when we factor trinomials of the type x 2 bx c. Find the missing numbers in the table.
Product Factor Factor Sum

40 5 8 13

63

36

72

140

96

48

168

110 9 10 24 18 18 3

16

20

38

4

4

14

29

21

122. A factored polynomial for the shaded area in this rectangle is A B A B .
A A B B

A A B B

a) Find a polynomial for the area of the entire rectangle. b) Find a polynomial for the sum of the areas of the two small unshaded rectangles. c) Find a polynomial for the area in part (a) minus the area in part (b). d) Find a polynomial for the area of the shaded region and compare this with the polynomial found in part (c).

Use the TABLE or GRAPH feature to check whether each of the following is correct. 123. x 1
2

x2

2x

1

124. x

2

2

x2

4x

4

ISBN:0-536-47742-6

125. x

3 x

3

x2

6

126. x

3 x

2

x2

x

6

285
Exercise Set 4.6
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Objectives
Evaluate a polynomial in several variables for given values of the variables. Identify the coefficients and the degrees of the terms of a polynomial and the degree of a polynomial. Collect like terms of a polynomial. Add polynomials. Subtract polynomials. Multiply polynomials.

4.7
3x xy 2

OPERATIONS WITH POLYNOMIALS IN SEVERAL VARIABLES

The polynomials that we have been studying have only one variable. A polynomial in several variables is an expression like those you have already seen, but with more than one variable. Here are two examples: 5y 4, 8xy 2z 2x 3z 13x 4y 2 15.

Evaluating Polynomials
EXAMPLE 1

Evaluate the polynomial 4 2 and y with 5: 8x 3y 3 4 4 3 6 8052. 2 50

3x

xy 2

8x 3y 3 when x

2

and y 4

5. 3x xy
2

We replace x with

2 8000

52

8

2

3

53

1. Evaluate the polynomial 4 when x 3x xy 2 2 and y 8x 3y 3 5.

Male Caloric Needs. The number of calories needed each day by a moderately active man who weighs w kilograms, is h centimeters tall, and is a years old can be estimated by the polynomial
EXAMPLE 2

19.18w

7h

9.52a

92.4.

2. Evaluate the polynomial 8xy 2 when x 2x 3z 1, y 13x 4y 2 5 4.

The author of this text is moderately active, weighs 87 kg, is 185 cm tall, and is 64 yr old. What are his daily caloric needs?
Source: Parker, M., She Does Math. Mathematical Association of America

3, and z

3. Female Caloric Needs. The number of calories needed each day by a moderately active woman who weighs w pounds, is h inches tall, and is a years old can be estimated by the polynomial 917 6w 6h 6a.

Christine is moderately active, weighs 125 lb, is 64 in. tall, and is 27 yr old. What are her daily caloric needs?
Source: Parker, M., She Does Math. Mathematical Association of America

We evaluate the polynomial for w 19.18w 7h 9.52a 92.4 19.18 87 2446.78. 7 185

87, h

185, and a 92.4

64:

9.52 64

Substituting
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His daily caloric need is about 2447 calories.

Answers on page A-18 286
CHAPTER 4: Polynomials: Operations

Do Exercises 1–3.

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Coefficients and Degrees
The degree of a term is the sum of the exponents of the variables. The degree of a polynomial is the degree of the term of highest degree.
EXAMPLE 3 Identify the coefficient and the degree of each term and the degree of the polynomial

4. Identify the coefficient of each term: 3xy 2 3x 2y 2y 3 xy 2.

9x 2y 3
TERM

14xy 2z 3

xy

4y

5x 2
DEGREE

7.
DEGREE OF THE POLYNOMIAL

COEFFICIENT

9x 2y 3 14xy 2z 3 xy 4y 5x 2 7

9 14 1 4 5 7

5 6 2 1 2 0

6 Think: 4y Think: 7 4y 1. 2 7x 0, or 7x 0y 0z 0.

5. Identify the degree of each term and the degree of the polynomial 4xy 2 7x 2y 3z 2 5x 2y 4.

Do Exercises 4 and 5.

Collecting Like Terms
Like terms have exactly the same variables with exactly the same exponents. For example, 3x 2y 3 and But 13xy 5 and 2x 2y 5 are not like terms, because the x-factors have different exponents; and 3xyz 2 and 4xy are not like terms, because there is no factor of z 2 in the second expression. Collecting like terms is based on the distributive laws.
EXAMPLES

7x 2y 3 are like terms;
Collect like terms. 6. 4x 2y 3xy 2x 2y

9x 4z 7 and 12x 4z 7 are like terms.

Collect like terms. 5x 2y 7b 2 3xy 2
2

4. 5x 2y 5. 8a 2 6. 7xy

3xy 2 2ab 5xy 2 2xy

xy 2 4a 2 7 5x

5 9ab 6x 3
3

5 x 2y 17b 2 9xy

3 12a 2

1 xy 2 11ab y 1

2xy 2 10b 2

7.

3pq 5pqr 3

5pqr 3 4

12

8pq

11x 3

16xy

y

8

Do Exercises 6 and 7.
ISBN:0-536-47742-6

Answers on page A-18 287
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.7 Operations with Polynomials in Several Variables

Add. 8. 4x 3 4x 2 8y 3 8x 3 2x 2 4y 5

Addition
We can find the sum of two polynomials in several variables by writing a plus sign between them and then collecting like terms.
EXAMPLE 7

Add: 3y 5 3x 3

5x 3 5y 2 8 x3 4x 2 3y

3y 8x 3 4x 2 2y 2

5y 2 4x 2 3y 5

8x 3 7y 2

4x 2

7y 2 .

5x 3

7 y2

9. 13x 3y 3x 2y 5y x 3y 4x 2y 3xy 3y

EXAMPLE 8

Add: 4x 2y 5x 3 2 3xy 2 2x 2y 3x 3y 5.

5xy

2

We first look for like terms. They are 5xy 2 and 3xy 2, 4x 2y and 2x 2y, and 2 and 5. We collect these. Since there are no more like terms, the answer is 8xy 2 6x 2y 5x 3 3x 3y 3.

Do Exercises 8–10. 10. 5p 2q 4 2p 2q 2 6pq 2 3p 2q 5 3q

Subtraction
We subtract a polynomial by adding its opposite, or additive inverse. The opposite of the polynomial 4x 2y 6x 3y 2 x 2y 2 5y is 4x 2y
EXAMPLE 9

6x 3y 2 Subtract: x 3y 2

x 2y 2

5y

4x 2y

6x 3y 2

x 2y 2

5y.

4x 2y
Subtract. 11. 4s 4t s 3t 2 4s 4t 5s 3t 2 2s 2t 3 s 2t 2

3x 2y 3

6y

10

4x 2y

6x 3y 2

x 2y 2

5y

8.

We have 4x 2y x 3y 2 4x y 7x 3y 2
2

3x 2y 3 x y
3 2

6y 3x y
2 3

10 6y 11y

4x 2y 10 18.

6x 3y 2
2

x 2y 2
3 2

5y x y
2 2

8 5y 8

4x y

6x y

Finding the opposite by changing the sign of each term

3x 2y 3

x 2y 2

Collecting like terms. (Try to write just the answer!)

Caution!
12. 5p 4q 5p 3q 2 3p 2q 3 4 2 4p 4q 4p 3q 2 7q 2 3 4 p q 2q 7 Do not add exponents when collecting like terms—that is, 7x 3 7x
3

8x 3 8x
3

15x 6; 15x .
3

Wrong Correct

Do Exercises 11 and 12.

ISBN:0-536-47742-6

Answers on page A-18 288
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Multiplication
To multiply polynomials in several variables, we can multiply each term of one by every term of the other. We can use columns for long multiplications as with polynomials in one variable. We multiply each term at the top by every term at the bottom. We write like terms in columns, and then we add.
EXAMPLE 10

Multiply. 13. x 2y 3 2x x 3y 2 3x

14. p 4q

2p 3q 2

3q 3

p

2q

Multiply: 3x 2y 3x y 6x 2y 2 2x 2y 2 4x 2y 2
2

2xy 3y 2y

3y xy

2y .

2xy xy 4xy 2 3xy 2 xy 2

6y 2 6y 2

Multiplying by 2y Multiplying by xy Adding

Multiply. 15. 3xy 2x x 2 2xy 2

3x y

3 2

3x 3y 2

Do Exercises 13 and 14. 16. x 3y 2x 5y

Where appropriate, we use the special products that we have learned.
EXAMPLES

Multiply. y2 3q x 3y 3 2p 2 2p 2 x 2y 3 3pq 7pq B 2x 2y 2 2xy 2
Using FOIL 17. 4x 5y
2

11. x 2y 12. p

2x xy 2 5q 2p

10pq 15q 2 B2 2y B
2

15q 2

A 13. 3x A 14. 2y 2

B 2y

2 2

A2 3x
2 2 2

2 A

18. 3x 2

2xy 2

2

2 3x 2y 2
2

9x 2 B2 5x 2y
2

12xy

4y 2

B

A2 2y 2 4y 4

A

5x 2y

2 2y 2 5x 2y 20x 2y 3 25x 4y 2 B2
2

19. 2xy 2

3x 2xy 2

3x

A 15. 3x 2y 16. 2x 3y 2

B

A

B 2y 5t

A2 3x 2y

2y 3x 2y

2y

2

9x 4y 2

4y 2

20. 3xy 2

4y

3xy 2

4y

5t 2x 3y 2

5t 5t
2

2x 3y 2 5t 2x 3y 2 A2
2

2x 3y 2 25t 2 4x 6y 4
21. 3y 4 3x 3y 4 3x

A 17. 2x 3

B 2y 2x

A 3

B 2y 2x 4x 2

B2 3
2

2y 9

2

12x

4y 2
22. 2a 5b c 2a 5b c

ISBN:0-536-47742-6

Remember that FOIL will always work when you are multiplying binomials. You can use it instead of the rules for special products, but those rules will make your work go faster.
Do Exercises 15–22.

Answers on page A-18 289

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.7 Operations with Polynomials in Several Variables

4.7
1. x 2 y2 xy 5. 8xyz

EXERCISE SET
3, y y2

For Extra Help
2, and z xy 5. 3. x 2

MathXL

MyMathLab

InterAct Math

Math Tutor Digital Video Center Tutor CD 2 Videotape 4

Student’s Solutions Manual

Evaluate the polynomial when x 2. x 2

3y 2

2xy

4. x 2

4xy

5y 2

6.

3xyz 2

7. xyz 2

z

8. xy

xz

yz

Lung Capacity. The polynomial equation C 0.041h 0.018A 2.69 can be used to estimate the lung capacity C, in liters, of a female of height h, in centimeters, and age A, in years. Use this formula for Exercises 9 and 10.

9. Find the lung capacity of a 20-year-old woman who is 165 cm tall.

10. Find the lung capacity of a 50-year-old woman who is 160 cm tall.

Altitude of a Launched Object. The altitude h, in meters, of a launched object is given by the polynomial equation h h0 vt 4.9t 2, where h0 is the height, in meters, from which the launch occurs, v is the initial upward speed (or velocity), in meters per second (m s), and t is the number of seconds for which the object is airborne. Use this formula for Exercises 11 and 12.
32 m

11. A model rocket is launched from the top of the Lands End Arch, near San Lucas, Baja, Mexico, 32 m above the ground. The upward speed is 40 m s. How high will the rocket be 2 sec after the blastoff?

12. A golf ball is thrown upward with an initial speed of 30 m s by a golfer atop the Washington Monument, which is 160 m above the ground. How high above the ground will the ball be after 3 sec?

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290
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Surface Area of a Right Circular Cylinder. The surface area S of a right circular cylinder is given by the polynomial equation S 2 rh 2 r 2, where h is the height and r is the radius of the base. Use this formula for Exercises 13 and 14.

r

h

13. A 12-oz beverage can has a height of 4.7 in. and a radius of 1.2 in. Evaluate the polynomial when h 4.7 and r 1.2 to find the area of the can. Use 3.14 for .

14. A 26-oz coffee can has a height of 6.5 in. and a radius of 2.5 in. Evaluate the polynomial when h 6.5 and r 2.5 to find the area of the can. Use 3.14 for .

Surface Area of a Silo. A silo is a structure that is shaped like a right circular cylinder with a half sphere on top. The surface area S of a silo of height h and radius r (including the area of the base) is given by the polynomial equation S 2 rh r 2. Note that h is the height of the entire silo. 15. A container of tennis balls is silo-shaped, with a height of 7 1 in. and a radius of 1 1 in. 2 4 Find the surface area of the container. Use 3.14 for . 16. A 1 1 -oz bottle of roll-on deodorant has a height of 4 in. 2 and a radius of 3 in. Find the surface area of the bottle 4 if the bottle is shaped like a silo. Use 3.14 for .

r

h

Identify the coefficient and the degree of each term of the polynomial. Then find the degree of the polynomial. 17. x 3y 2xy 3x 2 5 18. 5y 3 y2 15y 1

19. 17x 2y 3

3x 3yz

7

20. 6

xy

8x 2y 2

y5

Collect like terms. 21. a b 2a 3b 22. y 2 1 y 6 y2

23. 3x 2y

2xy 2

x2

24. m 3

2m 2n

3m 2

3mn 2

25. 6au
ISBN:0-536-47742-6

3av

14au

7av

26. 3x 2y

2z 2y

3xy 2

5z 2y

27. 2u2v

3uv 2

6u2v

2uv 2

28. 3x 2

6xy

3y 2

5x 2

10xy

5y 2

291
Exercise Set 4.7
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Add. 29. 2x 2 xy y2 x2 3xy 2y 2 30. 2zt z2 5t 2 z2 3zt t2

31. r

2s

3

2r

s

s

4

32. ab

2a

3b

5a

4b

3a

7ab

8b

33. b 3a 2

2b 2a 3

3ba

4

b 2a 3

4b 3a 2

2ba

1

34. 2x 2

3xy y 2 x xy y 2
2

4x 2

6xy

y2

Subtract. 35. a 3 b3 a 2b ab 2 b3 a3 36. x 3 y3 2x 3 x 2y xy 2 2y 3

37. xy

ab

8

xy

3ab

6

38. 3y 4x 2 2y 3x 3y 7 2y 4x 2 2y 3x 4y

2x

5

39.

2a

7b

c

3b

4c

8d

40. Subtract 5a

2b from the sum of 2a

b and 3a

b.

Multiply. 41. 3z u 2z 3u 42. a b a2 b2 2ab 43. a 2b 2 a 2b 5

44. xy

7 xy

4

45. a 3

bc a 3

bc

46. m 2

n2

mn m 2

mn

n2

47.

y 4x

y2

1

y2

1

48. a

b a2

ab

b2

49. 3xy

1 4xy

2

Copyright © 2007 Pearson Education, Inc.

ISBN:0-536-47742-6

50. m 3n

8 m 3n

6

51. 3

c 2d 2 4

c 2d 2

52. 6x

2y 5x

3y

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Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

53. m 2

n2 m

n

54. pq

0.2 0.4pq

0.1

55. xy

x 5y 5 x 4y 4

xy

56. x

y 3 2y 3

x

57. x

h

2

58. 3a

2b

2

59. r 3t 2

4

2

60. 3a 2b

b2

2

61.

p4

m 2n 2

2

62. 2ab

cd

2

63. 2a 3

1 3 2 2b

64.

3x x

8y

2

65. 3a a

2b

2

66. a 2

b

2

2

67. 2a

b 2a

b

68. x

y x

y

69. c 2

d c2

d

70.

p3

5q

p3

5q

71. ab

cd 2 ab

cd 2

72. xy

pq xy

pq

73. x

y

3 x

y

3

74.

p

q

4

p

q

4

75. x

y

z x

y

z

76. a

b

c a

b

c

77. a

b

c a

b

c

78. 3x

2

5y 3x

2

5y

79. x 2

4y

2 3x 2

5y

3

80. 2x 2

7y

4 x2

y

3

81.
ISBN:0-536-47742-6

DW Is it possible for a polynomial in four variables to have a degree less than 4? Why or why not?

82.

DW Can the sum of two trinomials in several variables be a trinomial in one variable? Why or why not?

293
Exercise Set 4.7
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

SKILL MAINTENANCE
In which quadrant is the point located? [3.1a] 83. 2, 5 84. 8, 9 85. 16, 23 86. 3, 2

Graph. [3.3b] 87. 2x 10 88. y 4 89. 8y 16 0 90. x 4

SYNTHESIS
Find a polynomial for the shaded area. (Leave results in terms of 91. y x

where appropriate.) 93. 94. x b b b b a

92. a x

y

x

b

y Hint: These are semicircles.

b b a b

b

Find a formula for the surface area of the solid object. Leave results in terms of . 95. n m h

96. h r x

x

97. Observatory Paint Costs. The observatory at Danville University is shaped like a silo that is 40 ft high and 30 ft wide (see Exercise 15). The Heavenly Bodies Astronomy Club is to paint the exterior of the observatory using paint that covers 250 ft 2 per gallon. How many gallons should they purchase?
30 ft

98. Interest Compounded Annually. An amount of money P that is invested at the yearly interest rate r grows to the amount P1 r t after t years. Find a polynomial that can be used to determine the amount to which P will grow after 2 yr.

40 ft

99. Suppose that $10,400 is invested at 8.5% compounded annually. How much is in the account at the end of 5 yr? (See Exercise 98.)

100. Multiply: x

a x

b x

a x

b.

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ISBN:0-536-47742-6

294
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.8

DIVISION OF POLYNOMIALS

Objectives
Divide a polynomial by a monomial. Divide a polynomial by a divisor that is a binomial.

In this section, we consider division of polynomials. You will see that such division is similar to what is done in arithmetic.

Dividing by a Monomial
We first consider division by a monomial. When dividing a monomial by a monomial, we use the quotient rule of Section 4.1 to subtract exponents when the bases are the same. We also divide the coefficients.
EXAMPLES Divide. 1. 20x 3 5x

Divide. x2 x 1 3 18 3 5x 2 x9 x2 x 10 x3
1

1.

10x 2 2x

10 2 1x 9 3x 2

5x
2

Caution!
1 7 x 3
3

x9 2. 3x 2 3.

1 9 x 3

18x 10 3x 3

The coefficients are divided but the exponents are subtracted. 28x 14 4x 3

6x 10 14a

6x 7 b 14ab
3

42a 2b 5 4. 3ab 2

42 a 2 b 5 3 a b2

2 1 5 2

2.

Do Exercises 1–4.

To divide a polynomial by a monomial, we note that since A C B C A C B ,

it follows that A C B A C B . C
Switching the left and right sides of the equation 3. 56p 5q 7 2p 2q 6

This is actually the procedure we use when performing divisions like 86 Although we might write 86 2 43,

2.

we could also calculate as follows: 86 2 80 2 6 80 2 6 2 40 3 43.
4. x5 4x

Similarly, to divide a polynomial by a monomial, we divide each term by the monomial.
EXAMPLE 5

Divide: 9x 8

12x 6 9x 8 9x 8 3x 2

3x 2 . 12x 6

We have
ISBN:0-536-47742-6

9x 8

12x 6

3x 2

3x 2 12x 6 . 3x 2
To see this, add and get the original expression.

Answers on page A-19 295
4.8 Division of Polynomials

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Divide. Check the result. 5. 28x 7 32x 5 4x 3

We now perform the separate divisions: 9x 8 3x 2 12x 6 3x 2 9 3 3x 8 3x 6 x8 x2
2

12 x 6 3 x2 4x 6
2

Caution!
The coefficients are divided, but the exponents are subtracted.

4x 4.

To check, we multiply the quotient 3x 6 3x 2 3x 6
6. 2x 3 6x 2 4x 2x

4x 4 by the divisor 3x 2: 3x 2 4x 4 9x 8 12x 6. 4x 4.

4x 4

3x 2 3x 6

This is the polynomial that was being divided, so our answer is 3x 6
Do Exercises 5–7. EXAMPLE 6

Divide and check: 10a 5b 4 2a 3b 2 2a 2b 6a 2b 10a 5b 4 2a 2b 10 5 2 4 a b 2

2a 3b 2 2a 3b 2 2a 2b
1

6a 2b 6a 2b 2a 2b
1

2a 2b .

10a 5b 4

2 3 2 2 a b 2 3

6 2

7. 6x 2

3x

2

3

5a 3b 3 Check: 2a 2b 5a 3b 3 Our answer, 5a 3b 3 ab ab 3

ab

2a 2b 5a 3b 3 10a 5b 4

2a 2b ab 6a 2b

2a 2b 3

2a 3b 2

3, checks.

Divide and check. 8. 8x 2 3x 1 2

To divide a polynomial by a monomial, divide each term by the monomial.

Do Exercises 8 and 9.

Dividing by a Binomial
Let’s first consider long division as it is performed in arithmetic. When we divide, we repeat the following procedure. To carry out long division: 1. 2. 3. 4. Divide, Multiply, Subtract, and Bring down the next term.

9.

2x 4y 6

3x 3y 4 x 2y 2

5x 2y 3

We review this by considering the division 3711 8 4 3 7 1 1 3 2 5 1 1 Divide: 37 2 Multiply: 4 3 Subtract: 37 8 8 32 4. 32. 5.

8. 8

Answers on page A-19 296
CHAPTER 4: Polynomials: Operations

4 3 7 3 2 5 4

6 3 1 1 1 8 3 1 2 4 7

ISBN:0-536-47742-6

4 Bring down the 1.

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Next, we repeat the process two more times. We obtain the complete division as shown on the right above. The quotient is 463. The remainder is 7, expressed as R 7. We write the answer as 463 R 7 or 463 7 8 463 7 . 8

10. Divide and check: x2 x 6 x 3.

We check by multiplying the quotient, 463, by the divisor, 8, and adding the remainder, 7: 8 463 7 3704 7 3711.

Now let’s look at long division with polynomials. We use this procedure when the divisor is not a monomial. We write polynomials in descending order and then write in missing terms.
EXAMPLE 7

Divide x 2 6

5x

6 by x

2.
x.

x

x 2 x2 x2

5x 2x 3x

Divide the first term by the first term: x 2 x Ignore the term 2. Multiply x above by the divisor, x 2. Subtract: x 2 5x x2 2x x2 3x. 5x

x2

2x

We now “bring down” the next term of the dividend—in this case, 6. x x 2 x2 x2 3 5x 2x 3x 3x
Divide the first term by the first term: 3x x 3.

6 6 6 0
The 6 has been “brought down.” Multiply 3 by the divisor, x 2. Subtract: 3x 6 3x 6 3x 6 3x 6 0.

The quotient is x 3. The remainder is 0, expressed as R 0. A remainder of 0 is generally not listed in an answer. To check, we multiply the quotient by the divisor and add the remainder, if any, to see if we get the dividend: Divisor Quotient Remainder Dividend

⎧ ⎨ ⎩ x 2 x x 3 x2 x2 x
ISBN:0-536-47742-6

Do Exercise 10. EXAMPLE 8

We now “bring down” the next term of the dividend—in this case, x 3 x2 x2 5 2x 3x 5x 5x
Divide the first term by the first term: 5x x

Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

⎧ ⎨ ⎩ x 3 2x 3x 5x 12 12 12 15 3

⎧ ⎪ ⎨ ⎪ ⎩

}

0

x

2

5x

6.

The division checks.

Divide and check: x 2

2x

12

x

3.

Divide the first term by the first term: x 2 x Multiply x above by the divisor, x Subtract: x
2

Answer on page A-19
x.

3. x2 5x. 2x x2 3x

2x

x

2

3x

Study Tips
FORMING A STUDY GROUP
Consider forming a study group with some of your fellow students. Exchange e-mail addresses, telephone numbers, and schedules so that you can coordinate study time for homework and tests.

12.
5.

Bring down the 12. Multiply 5 above by the divisor, x Subtract: 5x 12 5x 15 5x 3. 12

3. 5x 15

297
4.8 Division of Polynomials

Divide and check. 11. x 2 x2 2x 8

The answer is x
Quotient

5 with R
⎧ ⎨ ⎩

3, or 3 x 3 .
Remainder Divisor

x

5

(This is the way answers will be given at the back of the book.) Check: We can check by multiplying the divisor by the quotient and adding the remainder, as follows: x 3 x 5 3 x2 x
2

⎧ ⎨ ⎩

2x 2x

15 12.

3

12. x

3 x2

7x

10

When dividing, an answer may “come out even” (that is, have a remainder of 0, as in Example 7), or it may not (as in Example 8). If a remainder is not 0, we continue dividing until the degree of the remainder is less than the degree of the divisor. Check this in each of Examples 7 and 8.
Do Exercises 11 and 12. EXAMPLE 9

Divide and check: x 3 x 0x 2 x2 x2 x2 1 0x 0x x x x 1

1

x

1.

x

x2 1 x3 x3

Fill in the missing terms (see Section 4.3). Subtract: x 3 x3 x2 x 2. Subtract: x2 1 x2 x x 1 x. 0.

Divide and check. 13. x 3 1 x 1

1 1 0

Subtract: x

The answer is x 2
EXAMPLE 10

x

1. The check is left to the student. 7x 2 4x 13 3x 1.

Divide and check: 9x 4 13

3x

3x 3 1 9x 4 9x 4

14. 8x 4

10x 2

2x

9

4x

2

x 2 2x 2 0x 3 7x 2 4x 3x 3 3x 3 7x 2 3x 3 x2 6x 2 4x 6x 2 2x 6x 6x x2 2x 2 x2 x 7x
2 2

Fill in the missing term. Subtract: 9x 4 9x 4 3x 3 Subtract: 3x 3 7x 2

3x 3.

3x 3 6x 2 6x

x2 2x 2

6x 2. 6x. 11.

13 2 11 2 with R

Subtract: 6x 2 4x Subtract: 6x 13

The answer is 3x 3 3x 3 Check: 3x x2

2x

11; or

11 . 3x 1 2x 2x 4x 2 2 13 11 9x 4 3x 3 6x 2 6x 11
ISBN:0-536-47742-6

1 3x 3 3x 9x
4 3

Answers on page A-19
Do Exercises 13 and 14.

298
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4.8
1. 24x 4 8

EXERCISE SET

For Extra Help

MathXL

MyMathLab

InterAct Math

Math Tutor Digital Video Center Tutor CD 2 Videotape 4

Student’s Solutions Manual

Divide and check. 2. 2u2 u 3. 25x 3 5x 2

4.

16x 7 2x 2

5.

54x 11 3x 8

6.

75a 10 3a 2

7.

64a 5b 4 16a 2b 3

8.

34p 10q 11 17pq 9

9.

24x 4

4x 3 8

x2

16

10.

12a 4

3a 2 6

a

6

11.

u

2u2 u

u5

12.

50x 5

7x 4 x

x2

13. 15t 3

24t 2

6t

3t

14. 25t 3

15t 2

30t

5t

15. 20x 6

20x 4

5x 2

5x 2

16. 24x 6

32x 5

8x 2

8x 2

17. 24x 5

40x 4

6x 3

4x 3

18. 18x 6

27x 5

3x 3

9x 3

19.

18x 2

5x 2

2

20.

15x 2

30x 3

6

21.

12x 3

26x 2 2x

8x

22.
ISBN:0-536-47742-6

2x 4

3x 3 x2

5x 2

23.

9r 2s 2

3r 2s 3rs

6rs 2

24.

4x 4y

8x 6y 2 4x 4y

12x 8y 6

299
Exercise Set 4.8
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Divide. 25. x 2 4x 4 x 2 26. x 2 6x 9 x 3 27. x 2 10x 25 x 5

28. x 2

8x

16

x

4

29. x 2

4x

14

x

6

30. x 2

5x

9

x

2

31.

x2 x

9 3

32.

x2 x

25 5

33.

x5 x

1 1

34.

x4 x

81 3

35.

8x 3

22x 2 5x 4x 3

12

36.

2x 3

9x 2 2x

11x 3

3

37. x 6

13x 3

42

x3

7

38. x 6

5x 3

24

x3

3

39. 15x 3

8x 2

11x

12

5x

1

40. 20x 4

2x 3

5x

3

2x

3

41. t 3

t2

t

1

t

1

42. t 3

t2

t

1

t

1

43.

DW How is the distributive law used when dividing a polynomial by a binomial?

44.

DW On an assignment, Emma incorrectly writes
4x 2 6x.
Copyright © 2007 Pearson Education, Inc.

12x 3 6x 3x

What mistake do you think she is making and how might you convince her that a mistake has been made?

ISBN:0-536-47742-6

300
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

SKILL MAINTENANCE

i

VOCABULARY REINFORCEMENT

In each of Exercises 45–52, fill in the blank with the correct term from the given list. Some of the choices may not be used. 45. The rule asserts that when multiplying with exponential notation, if the bases are the same, keep the base and add the exponents. [4.1d] 46. A(n) is an expression of the type ax n, where a is a real-number constant and n is a nonnegative integer. [4.3a, i] 47. The principle asserts that when we multiply or divide by the same nonzero number on each side of an equation, we get equations. [2.2a] 48. Vertical lines are graphs of equations of the type . [3.3b] 49. A(n) such as 5x 4 7x 2 is a polynomial with three terms, 4. [4.3i] x y a b

slope positive absolute value equivalent inverse quotient product monomial binomial trinomial addition multiplication

50. The rule asserts that when dividing with exponential notation, if the bases are the same, keep the base and subtract the exponent of the denominator from the exponent of the numerator. [4.1e] 51. The of a number is its distance from zero on a number line. [1.2e] 52. The of the line y mx b is m. [3.4c]

SYNTHESIS
Divide. 53. x 4 9x 2 20 x2 4 54. y 4 a2 y a

55. 5a 3

8a 2

23a

1

5a 2

7a

2

56. 15y 3

30y

7

19y 2

3y 2

2

5y

57. 6x 5

13x 3

5x

3

4x 2

3x 4

3x 3

2x

1

58. 5x 7

3x 4

2x 2

10x

2

x2

x

1

59. a 6

b6

a

b

60. x 5

y5

x

y

ISBN:0-536-47742-6

If the remainder is 0 when one polynomial is divided by another, the divisor is a factor of the dividend. Find the value(s) of c for which x 1 is a factor of the polynomial. 61. x 2 4x c 62. 2x 2 3cx 8 63. c 2x 2 2cx 1

301
Exercise Set 4.8
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4

Summary and Review

The review that follows is meant to prepare you for a chapter exam. It consists of three parts. The first part, Concept Reinforcement, is designed to increase understanding of the concepts through true/false exercises. The second part is a list of important properties and formulas. The third part is the Review Exercises. These provide practice exercises for the exam, together with references to section objectives so you can go back and review. Before beginning, stop and look back over the skills you have obtained. What skills in mathematics do you have now that you did not have before studying this chapter?

i CONCEPT REINFORCEMENT
Determine whether the statement is true or false. Answers are given at the back of the book. 1. If one of the terms in an algebraic expression has a power in it, then the expression is a polynomial. 2. All trinomials are polynomials. 3. x 2 x 3 4. x y
2

x6 x2 y2

5. The square of the difference of two expressions is the difference of the squares of the two expressions 6. The product of the sum and the difference of two expressions is the difference of the squares of the expressions.

IMPORTANT PROPERTIES AND FORMULAS
FOIL: Square of a Sum: Square of a Difference: Product of a Sum and a Difference: A A A A B C B A B A B A D B B B AC A A A2 AD B B
2 2

BC A2 A2

BD 2AB 2AB B2 B2

B2

Definitions and Rules for Exponents: See p. 239.

Review Exercises
Multiply and simplify. [4.1d, f] 1. 7
2

Simplify. 2. y
7

7
5

4

y

3

y

8. 3t 4

2

[4.2a, b]

3. 3x

3x

9

4. t 8 t 0

9. 2x 3

2

3x

2

[4.1d], [4.2a, b]
ISBN:0-536-47742-6

Divide and simplify. [4.1e, f] 45 5. 2 4 a5 6. 8 a 7x 7. 7x
4 4

10.

2x y

3

[4.2b]

302
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

11. Express using a negative exponent:

1 . [4.1f] t5
4

Subtract. [4.4c] 29. 5x 2 4x 1 3x 2 1

12. Express using a positive exponent: y

. [4.1f] 30. 3x 5 4x 4 3x 2 3 2x 5 4x 4 3x 3 4x 2 5

13. Convert to scientific notation: 0.0000328. [4.2c] 14. Convert to decimal notation: 8.3 106. [4.2c] 31. Find a polynomial for the perimeter and for the area. [4.4d], [4.5b] w 3 w

Multiply or divide and write scientific notation for the result. [4.2d] 15. 3.8 104 5.5 10
1

16.

1.28 2.5

10 10

8 4

17. Diet-Drink Consumption. It has been estimated that there will be 292 million people in the United States by 2005 and that on average, each of them will drink 15.3 gal of diet drinks that year. How many gallons of diet drinks will be consumed by the entire population in 2005? Express the answer in scientific notation. [4.2e]
Source: U.S. Department of Agriculture

32. Find two algebraic expressions for the area of this figure. First, regard the figure as one large rectangle, and then regard the figure as a sum of four smaller rectangles. [4.4d]
3

18. Evaluate the polynomial x [4.3a]

2

3x

6 when x

1. t 19. Identify the terms of the polynomial 4y 5 7y 2 3y 2. [4.3b] 20. Identify the missing terms in x 3 x. [4.3h] Multiply. 33. x
2 3

t

4

21. Identify the degree of each term and the degree of the polynomial 4x 3 6x 2 5x 5 . [4.3g] 3 Classify the polynomial as a monomial, a binomial, a trinomial, or none of these. [4.3i] 22. 4x 23. 4 24. 7y 2 Collect like terms and then arrange in descending order. [4.3f] 25. 3x 2 26. x 2x
1 2

x

1 2

[4.6a]

34. 7x

1

2

[4.6c]

3

1 9t 3 7t 4 10t 2 35. 4x 2 5x 1 3x 2 [4.5d]

36. 3x 2

4 3x 2

4

[4.6b]

3 14x 4

5x 2 7x 2

1 1

x 4x 4

37. 5x 4 3x 3

8x 2

10x

2

[4.5b]

38. x

4 x

7

[4.6a]

Add. [4.4a] 27. 3x 4 x 3 x 4 5x 4 6x 2 x 28. 3x 5 4x 4 x 3 5x 5 5x 2 3 5x 4 x5 7x 3 3x 2 5 39. 3y 2 2y
2

[4.6c]

ISBN:0-536-47742-6

3x 4 2x 3

5x 3 5

3x 2

40. 2t 2

3 t2

7

[4.6a]

303
Summary and Review: Chapter 4
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

41. Evaluate the polynomial 2 when x 5xy y 1 and y
2

52.
3 6

DW Explain why the expression 578.6 scientific notation. [4.2c]

10

7

is not in

4xy x 2. [4.7a]

53. 42. Identify the coefficient and the degree of each term of the polynomial x 5y 7xy 9x 2 8. Then find the degree of the polynomial. [4.7b]

DW Write a short explanation of the difference between a monomial, a binomial, a trinomial, and a general polynomial. [4.3i]
SYNTHESIS

Find a polynomial for the shaded area. [4.4d], [4.6b] Collect like terms. [4.7c] 43. y w 2y 8w 5 x y

54.

44. m 6

2m 2n

m 2n 2

n 2m

6m 3

m 2n 2

7n 2m

x

y

45. Add: [4.7d] 5x 2 7xy y2 6x 2 3xy y2 x2 xy 2y 2 .

55. a a

20

46. Subtract: [4.7e] 6x 3y 2 4x 2y 6x 5x 3y 2 4x 2y 6x 2 6.

20

Multiply. [4.7f] 47. p q p2 pq q2 48. 3a 4
1 3 2 3b

56. Collect like terms: [4.1d], [4.2a], [4.3e] 3x 5 3x 3 x 6 2x
2

3x 4

2

2x 2

4

40x 2 x 3 2.

Divide. 49. 10x 3 x2 6x 2x [4.8a] 57. Solve: [2.3b], [4.6a] x 50. 6x 3 5x 2 13x 13 2x 3 [4.8b] 7 x 10 x 4 x 6.

58. The product of two polynomials is x 5 1. One of the polynomials is x 1. Find the other. [4.8b] 59. A rectangular garden is twice as long as it is wide and is surrounded by a sidewalk that is 4 ft wide (see the figure below). The area of the sidewalk is 256 ft 2. Find the dimensions of the garden. [2.3b], [4.4d], [4.5a], [4.6a]

51. The graph of the polynomial equation y 10x 3 10x is shown below. Use only the graph to estimate the value of the polynomial when x 1, x 0.5, x 0.5, and x 1. [4.3a] y 5 4

y

10x 3

10x

1 5 4 3 2 1 2 3 4 5 2 3 4 5

x 4 ft
ISBN:0-536-47742-6

304
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

4
1. 6
2

Chapter Test
3

For Extra Help

Work It Out! Chapter Test Video on CD

Multiply and simplify. 6 2. x 6 x 2 x 3. 4a
3

4a

8

Divide and simplify. 4. 35 32 5. x3 x8 6. 2x 2x
5 5

Simplify. 7. x 3
2

8.
3

3y 2
3

3

9. 2a 3b 2x 5
3

4

10.
4

ab c
2

3

11. 3x 2

2x 5

3

12. 3 x 2

13. 2x 2

3x 2

14. 2x

3x 2

4

15. Express using a positive exponent: 5 3.

16. Express using a negative exponent: 1 . y8 18. Convert to decimal notation: 5 10 8.

17. Convert to scientific notation: 3,900,000,000. Multiply or divide and write scientific notation for the answer. 19. 5.6 106 3.2 10 11

20. 2.4

105 5.4

1016

21. CD-ROM Memory. A CD-ROM can contain about 600 million pieces of information (bytes). How many sound files, each containing 40,000 bytes, can a CD-ROM hold? Express the answer in scientific notation. 23. Identify the coefficient of each term of the polynomial 1 5 x 7. 3x 25. Classify the polynomial 7 Collect like terms. 26. 4a 2 6 a2

22. Evaluate the polynomial x 5

5x

1 when x

2.

24. Identify the degree of each term and the degree of the polynomial 2x 3 4 5x 3x 6.

x as a monomial, a binomial, a trinomial, or none of these.

27. y 2

3y

y

3 2 y 4

28. Collect like terms and then arrange in descending order: 3
ISBN:0-536-47742-6

x2

2x 3

5x 2

6x

2x

x 5.

Add. 29. 3x 5 5x 3 5x 2 3 x 5 x 4 3x 3 3x 2 30. 2x 4 x4 2 x 3 5 4x 4 5x 2 1 x 3

305
Test: Chapter 4
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.

Subtract. 31. 2x 4 x3 8x 2 6x 3 6x 4 8x 2 2x 32. x 3 0.4x 2 12 x5 0.3x 3 0.4x 2 9

Multiply. 33. 3x 2 4x 2 3x 5 34. x 1 3
2

35. 3x

10 3x

10

36. 3b

5 b

3

37. x 6

4 x8

4

38. 8

y 6

5y

39. 2x

1 3x 2

5x

3

40. 5t

2

2

41. Collect like terms: x y
3

42. Subtract: xy
3

y

3

8

6x y

3

x y

2 2

11.

8a 2b 2

ab

b3

6ab 2

7ab

ab 3

5b 3 .

43. Multiply: 3x 5 Divide. 44. 12x 4 9x 3

4y 5 3x 5

4y 5 .

15x 2

3x 2

45. 6x 3

8x 2 y 5 4 3 2

14x

13

3x

2

46. The graph of the polynomial equation y x 3 5x 1 is shown at right. Use only the graph to estimate the value of the polynomial when x 1, x 0.5, x 0.5, x 1, and x 1.1.

5 4 3 2 1

1 2 3 4 5

1 2 3 4 5

x

y

x3

5x

1

47. Find a polynomial for the surface area of this right rectangular solid.

48. Find two algebraic expressions for the area of this figure. First, regard the figure as one large rectangle, and then regard the figure as a sum of four smaller rectangles. t 2

5 a

9 t
Copyright © 2007 Pearson Education, Inc.

2

SYNTHESIS
49. The height of a box is 1 less than its length, and the length is 2 more than its width. Find the volume in terms of the length. 50. Solve: x 5 x 5 x 6 2.

ISBN:0-536-47742-6

306
CHAPTER 4: Polynomials: Operations
Introductory and Intermediate Algebra, Third Edition, by Marvin L.Bittinger and Judith A.Beecher.Published by Addison Wesley. Copyright ©2007 by Pearson Education, Inc.