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297

Source: NASA.

5.1 Rules for Exponents 5.2 Addition and Subtrac-

tion of Polynomials 5.3 Multiplication of

Polynomials 5.4 Special Products 5.5 Integer Exponents and

the Quotient Rule 5.6 Division of Polynomials

Digital images were first sent between New York and London by cable in the early 1920s. Unfortunately, the transmission time was 3 hours and the quality was poor. Digital photography was developed further by NASA in the 1960s because ordinary pic- tures were subject to interference when transmitted through space. Today, digital pic- tures remain crystal clear even if they travel millions of miles. The following digital picture shows the planet Mars.

Whether they are taken with a webcam, with a smartphone, or by the Mars rover, digital images comprise tiny units called pixels, which are represented by numbers. As a result, mathematics plays an important role in digital images. In this chapter we illustrate some of the ways mathematics is used to describe digital pictures (see Example 4 and Exercise 80 in Section 5.4). We also discuss how mathematics is used to model things such as heart rate, computer sales, motion of the planets, and interest on money.

5 Polynomials and Exponents

If you want to do something, do it!

—PLAUTUS

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When evaluating expressions, evaluate exponents before performing addition, subtraction, multiplication, division, or negation.

5.1 Rules for Exponents Review of Bases and Exponents ● Zero Exponents ● The Product Rule ● Power Rules

A LOOK INTO MATH N Electronic devices such as tablet computers and smartphones store information as bits. A bit is either a 0 or a 1, and a string of 8 bits is called a byte. In the 1970s, IBM devel- oped punch cards made out of paper that could hold up to 120 bits of information. Today, many computer hard drives can hold more than 1 terabyte of information; that’s more than 8,000,000,000,000 bits! In mathematics, we often use exponents to express such large numbers. In this section, we discuss the rules for exponents.

Review of Bases and Exponents The expression 53 is an exponential expression with base 5 and exponent 3. Its value is

5 # 5 # 5 = 125. In general, b n is an exponential expression with base b and exponent n. If n is a natural number, it indicates the number of times the base b is to be multiplied with itself.

Exponent T

b n = b # b # b # g # b Base c n times

STUDY TIP

Exponents occur throughout mathematics. Because expo- nents are so important, this section is essential for your success in mathematics. It takes practice, so set aside some extra time.

EVALUATING EXPRESSIONS

When evaluating expressions, use the following order of operations.

1. Evaluate exponents. 2. Perform negation. 3. Do multiplication and division from left to right. 4. Do addition and subtraction from left to right.

EXAMPLE 1 Evaluating exponential expressions

Evaluate each expression.

(a) 1 + 24

4 (b) 3a1

3 b2 (c) - 24 (d) ( - 2)4

Solution (a) Evaluate the exponent first.

4 factors

1 + 24

4 = 1 +

2 # 2 # 2 # 2 4

= 1 + 16

4 = 1 + 4 = 5

∂

2 factors

(b) 3a1 3 b2 = 3a1

3 # 1

3 b = 3 # 1

9 =

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2995.1 RULES FOR EXPONENTS

4 factors

- 24 = - (2 # 2 # 2 # 2) = - 16.∂ 4 factors

(d) (�2)4 = (�2)(�2)(�2)(�2) = 16

Now Try Exercises 9, 11, 15, 17

NOTE: Parts (c) and (d) of Example 1 appear to be very similar. However, the negation sign is inside the parentheses in part (d), which means that the base for the exponential expression is - 2. In part (c), no parentheses are used, indicating that the base of the expo- nential expression is 2.

READING CHECK

• Explain how to tell the difference between a negative number raised to a power and the opposite of a positive number raised to a power.

TECHNOLOGY NOTE

Evaluating Exponents Exponents can often be evaluated on calculators by using the ^ key. The four expressions from Example 1 are evaluated with a calculator and the results are shown in the following two figures. When evaluating the last two expressions on your calculator, remember to use the negation key rather than the subtraction key.

1�2^4/4 5

3(1/3) 2�Frac 1/3

�2^4 �16

(�2)^4 16

CALCULATOR HELP To evaluate exponents, see Appendix A (page AP-1).

Zero Exponents So far we have discussed natural number exponents. What if an exponent is 0? What does 20 equal? To answer these questions, consider Table 5.1, which shows values for decreas- ing powers of 2. Note that each time the power of 2 decreases by 1, the resulting value is divided by 2. For this pattern to continue, we need to define 20 to be 1 because dividing 2 by 2 results in 1.

This discussion suggests that 20 = 1, and is generalized as follows.

TABLE 5.1 Powers of 2

Power of 2 Value

23 8

22 4

21 2

20 ? ZERO EXPONENT

For any nonzero real number b,

b 0 = 1.

The expression 00 is undefined.

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EXAMPLE 2 Evaluating zero exponents

Evaluate each expression. Assume that all variables represent nonzero numbers.

(a) 70 (b) 3a4 9 b0 (c) ax 2y5

3z b0

Solution (a) 70 = 1 (b) 314920 = 3(1) = 3. (Note that the exponent 0 does not apply to 3.) (c) All variables are nonzero, so the expression inside the parentheses is also nonzero.

Thus 1x 2y53z 20 = 1. Now Try Exercises 13, 41, 67

THE PRODUCT RULE

For any real number a and natural numbers m and n,

am # an = am + n.

READING CHECK

• State the product rule in your own words.

The expression 43 # 42 has 3 + 2 = 5 factors of 4, so the result is 43 + 2 = 45. To multiply exponential expressions with the same base, add exponents and keep the base.

5 factors

The Product Rule We can use a special rule to calculate products of exponential expressions provided their bases are the same. For example,

43 # 42 = (4 # 4 # 4) # (4 # 4) = 45.v

NOTE: The product 24 # 35 cannot be simplified by using the product rule because the exponential expressions have different bases: 2 and 3.

EXAMPLE 3 Using the product rule

Multiply and simplify. (a) 23 # 22 (b) x 4x 5 (c) 2 x 2 # 5x 6 (d) x 3(2 x + 3x 2) Solution (a) 23 # 22 = 23 + 2 = 25 = 32 (b) x 4x 5 = x 4 + 5 = x 9 (c) Begin by applying the commutative property of multiplication to write the product in a

more convenient order.

2 x 2 # 5x 6 = 2 # 5 # x 2 # x 6 = 10x 2 + 6 = 10x 8 (d) To simplify this expression, first apply the distributive property.

3�1 T T 3�2

x 3(2 x + 3x 2) = x 3 # 2 x + x 3 # 3x 2 = 2 x 4 + 3x 5 c Exponent is 1.

Now Try Exercises 21, 23, 27, 71

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3015.1 RULES FOR EXPONENTS

EXAMPLE 4 Applying the product rule

Multiply and simplify. (a) x # x 3 (b) (a + b)(a + b)4

Solution (a) Begin by writing x as x 1. Then x 1 # x 3 = x 1 + 3 = x 4. (b) First write (a + b) as (a + b)1. Then

(a � b)1 # (a � b)4 = (a � b)1 + 4 = (a � b)5. Now Try Exercises 19, 63

Power Rules How should (43)2 be evaluated? To answer this question, consider how the product rule can be used in evaluating

(43)2 = 43 # 43 = 43 + 3 = 46.

5 Product rule

5 3 + 3 = 3 # 2 Similarly,

5 + 5 + 5 = 5 # 3

$%& (a5)3 = a5 # a5 # a5 = a5 + 5 + 5 = a15.

Product rule

This discussion suggests that to raise a power to a power, we multiply the exponents.

RAISING A POWER TO A POWER

For any real number a and natural numbers m and n,

(am)n = amn.

EXAMPLE 5 Raising a power to a power

Simplify each expression. (a) (32)4 (b) (a3)2

Solution (a) (32)4 = 32

# 4 = 38 (b) (a3)2 = a3 # 2 = a6

Now Try Exercises 31, 33

To decide how to simplify the expression (2 x)3, consider

(2 x)3 = 2 x # 2 x # 2 x = (2 # 2 # 2) # (x # x # x) = 23x 3. 3 factors 3 factors 3 factors

∂ ¶ ¶

To raise a product to a power, we raise each factor to the power.

NOTE: If an exponent does not appear in an expression, it is assumed to be 1. For example, x can be written as x 1 and (x + y) can be written as (x + y)1.

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READING CHECK

• State the rule for raising a product to a power in your own words.

RAISING A PRODUCT TO A POWER

For any real numbers a and b and natural number n,

(ab)n = anb n.

EXAMPLE 6 Raising a product to a power

Simplify each expression. (a) (3z)2 (b) ( - 2 x 2)3 (c) 4(x 2y3)5 (d) ( - 22a5)3

Solution (a) (3z)2 = 32z2 = 9z2 (b) ( - 2 x 2)3 = ( - 2)3(x 2)3 = - 8x 6 (c) 4(x 2y3)5 = 4(x 2)5( y3)5 = 4 x 10y15 (d) ( - 22a5)3 = ( - 4a5)3 = ( - 4)3(a5)3 = - 64a15

Now Try Exercises 37, 39, 43, 45

The following equation illustrates another power rule.

a2 3 b4 = 2

3 # 2

3 =

2 # 2 # 2 # 2 3 # 3 # 3 # 3 =

4 factors

To raise a quotient to a power, raise both the numerator and the denominator to the power.

RAISING A QUOTIENT TO A POWER

For any real numbers a and b and natural number n,

aa b bn = an

b n . b � 0

EXAMPLE 7 Raising a quotient to a power

Simplify each expression.

(a) a2 3 b3 (b) aa

b b9 (c) aa + b

5 b2

Solution

(a) a2 3 b3 = 23

33 =

27 (b) aa

b b9 = a9

b 9

(c) Because the numerator is an expression with more than one term, we must place paren- theses around it before raising it to the power 2.

aa + b 5 b2 = (a + b)2

52 =

(a + b)2

Now Try Exercises 51, 53, 55

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3035.1 RULES FOR EXPONENTS

The five rules for exponents discussed in this section are summarized as follows.

MAKING CONNECTIONS

Raising a Sum or Difference to a Power

Although there are power rules for products and quotients, there are not similar rules for sums and differences. In general, (a + b)n 3 an + b n and (a - b)n 3 an - b n. For example, (3 + 4)2 = 72 = 49 but 32 + 42 = 9 + 16 = 25. Similarly, (4 - 1)3 = 33 = 27 but 43 - 13 = 64 - 1 = 63.

RULES FOR EXPONENTS

The following rules hold for real numbers a and b, and natural numbers m and n.

Description Rule Example

Zero Exponent b 0 = 1, for b � 0 ( - 13)0 = 1

The Product Rule am # an = am + n 54 # 53 = 54 + 3 = 57 Power to a Power (am)n = am # n ( y2)5 = y2 # 5 = y10

Product to a Power (ab)n = anb n ( pq)7 = p7q 7

Quotient to a Power aa b bn = an

b n , for b � 0 ax

y b3 = x 3

y3 , for y � 0

Simplification of some expressions may require the application of more than one rule of exponents. This is demonstrated in the next example.

EXAMPLE 8 Combining rules for exponents

Simplify each expression.

(a) (2a)2(3a)3 (b) aa2b 3 c b4 (c) (2 x 3y)2( - 4 x 2y3)3

Solution (a) (2a)2(3a)3 = 22a2 # 33a3 Raising a product to a power = 4 # 27 # a2 # a3 Evaluate powers; commutative property = 108a5 Product rule

(b) aa2b 3 c b4 = (a2)4(b 3)4

c 4 Raising a quotient to a power; raising

a product to a power

= a8b 12

c 4 Raising a power to a power

(c) (2 x 3y)2( - 4 x 2y3)3 = 22(x 3)2y2( - 4)3(x 2)3( y3)3 Raising a product to a power = 4 x 6y2( - 64)x 6y9 Raising a power to a power = 4( - 64)x 6x 6y2y9 Commutative property = - 256 x 12y11 Product rule

Now Try Exercises 47, 49, 61

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N REAL-WORLD CONNECTION Exponents occur frequently in calculations involving yearly percent increases, such as the increase in property value illustrated in the next example.

EXAMPLE 9 Calculating growth in property value

If a parcel of property increases in value by about 11% each year for 20 years, then its value will double three times. (a) Write an exponential expression that represents “doubling three times.” (b) If the property is initially worth $25,000, how much will it be worth after it doubles

3 times?

Solution (a) Doubling three times is represented by 23. (b) 23(25,000) = 8(25,000) = $200,000

Now Try Exercise 85

5.1 Putting It All Together

Bases and Exponents In the expression b n, b is the base and n is the exponent. If n is a natural number, then

b n = b # b # g # b. 5 n times

23 has base 2 and exponent 3.

91 = 9, 32 = 3 # 3 = 9, 43 = 4 # 4 # 4 = 64, and - 62 = - (6 # 6) = - 36

Zero Exponents For any nonzero number b, b 0 = 1. 50 = 1, x 0 = 1, and (xy3)0 = 1

The Product Rule For any real number a and natural numbers m and n,

am # an = am + n.

24 # 23 = 24 + 3 = 27, x # x 2 # x 6 = x 1 + 2 + 6 = x 9, and (x + 1) # (x + 1)2 = (x + 1)3

CONCEPT EXPLANATION EXAMPLES

Raising a Power to a Power

For any real number a and natural numbers m and n,

(am)n = amn.

(24)2 = 24 # 2 = 28,

(x 2)5 = x 2 # 5 = x 10, and

(a4)3 = a4 # 3 = a12

Raising a Product to a Power

For any real numbers a and b and natural number n,

(ab)n = anb n.

(3x)3 = 33x 3 = 27x 3, (x 2y)4 = (x 2)4y4 = x 8y4, and ( - xy)6 = ( - x)6y6 = x 6y6

Raising a Quotient to a Power

For any real numbers a and b and natural number n,

aa b bn = an

b n . b � 0

ax y b5 = x 5

y5 and

aa2b d 3 b4 = (a2)4b 4

(d 3)4 =

a8b 4

d 12

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3055.1 RULES FOR EXPONENTS

5.1 Exercises

CONCEPTS AND VOCABULARY

1. In the expression b n, b is the _____ and n is the _____.

2. The expression b 0 = for any nonzero number b.

3. am # an = 4. (am)n = 5. (ab)n = 6. aa

b bn =

PROPERTIES OF EXPONENTS

Exercises 7–18: Evaluate the expression.

7. 82 8. 43

9. ( - 2)3 10. ( - 3)4

11. - 23 12. - 34

13. 60 14. ( - 0.5)0

15. 3 + 42

2 16. 6 - a - 4

2 b2

17. 4a1 2 b3 18. 16a1

4 b2

Exercises 19–74: Simplify the expression. Assume that all variables represent nonzero numbers.

19. 3 # 32 20. 53 # 53 21. 42 # 46 22. 104 # 103

23. x 3x 6 24. a5a2

25. x 2x 2x 2 26. y7y3y0

27. 4 x 2 # 5x 5 28. - 2y6 # 5y2 29. 3( - xy3)(x 2y) 30. (a2b 3)( - ab 2)

31. (23)2 32. (103)4

33. (n3)4 34. (z7)3

35. x(x 3)2 36. (z3)2(5z5)

37. ( - 7b)2 38. ( - 4z)3

39. (ab)3 40. (xy)8

41. (2 x 2)0 42. (3a2)4

43. ( - 4b 2)3 44. ( - 3r 4t 3)2

45. (x 2y3)7 46. (rt 2)5

47. ( y3)2(x 4y)3 48. (ab 3)2(ab)3

49. (a2b)2(a2b 2)3 50. (x 3y)(x 2y4)2

51. a1 3 b3 52. a5

2 b2

53. aa b b5 54. ax

2 b4

55. ax - y 3 b3 56. a 4

x + y b2

57. a 5 a + b

b2 58. aa - b 2 b3

59. a2 x 5 b3 60. a3y

2 b4

61. a3x 2 5y4 b3 62. aa2b 3

3 b5

63. (x + y)(x + y)3 64. (a - b)2(a - b)

65. (a + b)2(a + b)3 66. (x - y)5(x - y)4

67. 6(x 4y6)0 68. axy z2 b0

69. a(a2 + 2b 2) 70. x 3(3x - 5y4) 71. 3a3(4a2 + 2b) 72. 2 x 2(5 - 4y3)

73. (r + t)(rt)

74. (x - y)(x 2y3)

75. Thinking Generally Students sometimes mistakenly apply the “rule” am # b n � (ab)m + n. In general, this equation is not true. Find values for a, b, m, and n with a � b and m � n that will make this equation true.

76. Thinking Generally Students sometimes mistakenly

apply the “rule” (a + b)n � an + b n. In general, this equation is not true. Find values for a, b, and n with a � b that will make this equation true.

APPLICATIONS

Exercises 77–80: Write a simplified expression for the area of the given figure.

77. 2x2

5x2

78. 2ab

2ab

79.

3x2

80.

4y2

7y3

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Exercises 81 and 82: Write a simplified expression for the volume of the given figure.

81.

82.

3a2

3a2 3a2

83. Compound Interest If P dollars are deposited in an account that pays 5% annual interest, then the amount of money in the account after 3 years is P(1 + 0.05)3. Find the amount when P = $1000.

84. Compound Interest If P dollars are deposited in an account that pays 9% annual interest, then the amount of money in the account after 4 years is P(1 + 0.09)4. Find the amount when P = $500.

85. Investment Growth If an investment increases in value by about 10% each year for 22 years, then its value will triple two times.

(a) Write an exponential expression that represents “tripling two times.”

(b) If the investment has an initial value of $8000, how much will it be worth if it triples two times?

86. Stock Value If a stock decreases in value by about 23% each year for 9 years, then its value will be halved three times. (a) Write an exponential expression that represents

“halved three times.” (b) If the stock is initially worth $88 per share, how

much will it be worth if it is halved three times?

WRITING ABOUT MATHEMATICS

87. Are the expressions (4 x)2 and 4 x 2 equal in value? Explain your answer.

88. Are the expressions 33 # 23 and 66 equal in value? Explain your answer.

5.2 Addition and Subtraction of Polynomials Monomials and Polynomials ● Addition of Polynomials ● Subtraction of Polynomials ● Evaluating Polynomial Expressions

A LOOK INTO MATH N If you have ever exercised strenuously and then taken your pulse immediately afterward, you may have discovered that your pulse slowed quickly at first and then gradually leveled off. A typical scatterplot of this phenomenon is shown in Figure 5.1(a). These data points cannot be modeled accurately with a line, so a new expression, called a polynomial, is needed to model them. A graph of a polynomial that models these data is shown in Figure 5.1(b) and discussed in Exercise 71. (Source: V. Thomas, Science and Sport.)

NEW VOCABULARY

n Monomial n Degree of a monomial n Coefficient of a monomial n Polynomial n Polynomial in one variable n Binomial n Trinomial n Degree of a polynomial n Like terms

Figure 5.1 Heart Rate After Exercising

10 2 3 4 5 6 7

100

125

150

175

200

Time (minutes)

(a)

P ul

se (

be at

s pe

r m

in ut

Heart Rate

10 2 3 4 5 6 7

100

125

150

175

200

Time (minutes)

(b)

P ul

se (

be at

s pe

r m

in ut

Polynomial Model

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3075.2 ADDITION AND SUBTRACTION OF POLYNOMIALS

Monomials and Polynomials A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers. Examples of monomials include

- 3, xy2, 5a2, - z 3, and - 1

2 xy3.

A monomial may contain more than one variable, but monomials do not contain division by variables. For example, the expression 3z is not a monomial. If an expression contains addi- tion or subtraction signs, it is not a monomial.

The degree of a monomial is the sum of the exponents of the variables. If the mono- mial has only one variable, its degree is the exponent of that variable. Remember, when a variable does not have a written exponent, the exponent is implied to be 1. A nonzero number has degree 0, and the number 0 has undefined degree. The number in a monomial is called the coefficient of the monomial. Table 5.2 contains the degree and coefficient of several monomials.

READING CHECK

• How do you determine the degree of a monomial?

A polynomial is a monomial or the sum of two or more monomials. Each monomial is called a term of the polynomial. Addition or subtraction signs separate terms. The expres- sion 2 x 2 - 3x + 5 is a polynomial in one variable with three terms. Examples of polyno- mials in one variable include

- 2 x, 3x + 1, 4y2 - y + 7, and x 5 - 3x 3 + x - 7.

These polynomials have 1, 2, 3, and 4 terms, respectively. A polynomial with two terms is called a binomial, and a polynomial with three terms is called a trinomial.

A polynomial can have more than one variable, as in

x 2y2, 2 xy2 + 5x 2y - 1, and a2 + 2ab + b 2.

Note that all variables in a polynomial are raised to natural number powers. The degree of a polynomial is the degree of the term (or monomial) with greatest degree.

READING CHECK

• How do you determine the degree of a polyomial?

TABLE 5.2 Properties of Monomials

Monomial - 5 6a3b - xy 7y3

Degree 0 4 2 3

Coe fficient - 5 6 - 1 7

EXAMPLE 1 Identifying properties of polynomials

Determine whether the expression is a polynomial. If it is, state how many terms and vari- ables the polynomial contains and give its degree.

(a) 7x 2 - 3x + 1 (b) 5x 3 - 3x 2y3 + xy2 - 2y3 (c) 4 x 2 + 5

x + 1 Solution (a) The expression 7x 2 - 3x + 1 is a polynomial with three terms and one variable. The

first term 7x 2 has degree 2 because the exponent on the variable is 2. The second term - 3x has degree 1 because the exponent on the variable is implied to be 1. The third term 1 has degree 0 because it is a nonzero number. The term with greatest degree is 7x 2, so the polynomial has degree 2.

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(b) The expression 5x 3 - 3x 2y3 + xy2 - 2y3 is a polynomial with four terms and two variables, x and y. The first term has degree 3 because the exponent on the variable is 3. The second term has degree 5 because the sum of the exponents on the variables x and y is 5. Likewise, the third term has degree 3 and the fourth term has degree 3. The term with greatest degree is - 3x 2y3, so the polynomial has degree 2 + 3 = 5.

Now Try Exercises 21, 23, 27

Figure 5.2 Adding lw + lw

l lw

+ l lw

= l 2lw

If two monomials contain the same variables raised to the same powers, we call them like terms. We can add or subtract (combine) like terms but cannot combine unlike terms. The terms lw and 2lw are like terms and can be combined geometrically, as shown in Figure 5.3. If we joined one of the small rectangles with area lw and a larger rect- angle with area 2 lw, then the total area is 3 lw.

The distributive property justifies combining like terms.

1lw + 2lw = (1 + 2)lw = 3lw

The rectangles shown in Figure 5.4 have areas of ab and xy. Together, their area is the sum, ab + xy. However, because these monomials are unlike terms, they cannot be combined into one term.

Figure 5.3 Adding lw + 2lw

l lw 3lw

+ l 2lw = l

2w 3w

STUDY TIP

Do you want to know what material will be covered on your next exam? Often, the best place to look is on previ- ously completed assignments and quizzes. If a topic is not discussed in class, is not found on the syllabus, and is not part of your assignments, then your time may be better spent studying other topics.

Figure 5.4 Unlike terms: ab + xy

a ab xy

+ x

xy x

= a

Addition of Polynomials Suppose that we have 2 identical rectangles with length l and width w, as illustrated in Figure 5.2. Then the area of one rectangle is lw and the total area is

lw + lw.

This area is equivalent to 2 times lw, which can be expressed as 2lw, or

lw + lw = 2lw.

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3095.2 ADDITION AND SUBTRACTION OF POLYNOMIALS

EXAMPLE 2 Adding like terms

State whether each pair of expressions contains like terms or unlike terms. If they are like terms, add them.

(a) 5x 2, - x 2 (b) 7a2b, 10ab 2 (c) 4rt 2, 1

2 rt 2

Solution (a) The terms 5x 2 and - x 2 have the same variable raised to the same power, so they are like

terms. To add like terms, add their coefficients. Note that the coefficient of - x 2 is - 1.

5x 2 + ( - x 2) = (5 + ( - 1))x 2 Distributive property = 4 x 2 Add.

(b) The terms 7a2b and 10 ab 2 have the same variables, but these variables are not raised to the same powers. They are unlike terms and cannot be added.

like terms. We add them as follows.

4rt 2 + 1 2

rt 2 = a4 + 1 2 brt 2 Distributive property

= 9

2 rt 2 Add.

Now Try Exercises 29, 31, 33

EXAMPLE 3 Adding polynomials

Add by combining like terms. (a) (3x + 4) + ( - 4 x + 2) (b) ( y2 - 2y + 1) + (3y2 + y + 11)

Solution (a) (3x + 4) + (�4 x + 2) = 3x + (�4 x) + 4 + 2 = (3 - 4)x + (4 + 2) = - x + 6

(b) ( y2 - 2y + 1) + (3y2 + y + 11) = y2 + 3y2 - 2y + y + 1 + 11 = (1 + 3)y2 + ( - 2 + 1)y + (1 + 11) = 4y2 - y + 12

Now Try Exercises 37, 39

Recall that the commutative and associative properties of addition allow us to rearrange a sum in any order. For example, if we write each subtraction in 2 x - 5 - 4 x + 10 as addition of the opposite, we have

2 x � 5 � 4 x + 10 = 2 x + (�5) + (�4 x) + 10,

and the terms can be rearranged as

2 x + (�4 x) + (�5) + 10 = 2 x � 4 x � 5 + 10.

If we pay attention to the sign in front of each term in a polynomial, the like terms can be combined without rearranging the terms, as demonstrated in the next example.

To add two polynomials, combine like terms, as illustrated in the next example.

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EXAMPLE 4 Adding polynomials

Add (x 3 - 3x 2 + 7x - 4) + (4 x 3 - 5x + 9) by combining like terms.

Solution Remove parentheses and identify like terms with their signs as shown.

x 3 � 3x 2 � 7x � 4 � 4 x 3 � 5x � 9

When like terms (of the same color) are added, the resulting sum is

5x 3 � 3x 2 � 2 x � 5.

Now Try Exercise 41

Polynomials can also be added vertically, as demonstrated in the next example.

EXAMPLE 5 Adding polynomials vertically

Simplify (3x 2 - 3x + 5) + ( - x 2 + x - 6).

Solution Write the polynomials in a vertical format and then add each column of like terms.

3x 2 - 3x + 5 - x 2 + x - 6 2 x 2 - 2 x - 1 Add like terms in each column.

Regardless of the method used, the same answer should be obtained. However, adding ver- tically requires that like terms be placed in the same column.

Now Try Exercise 47

Subtraction of Polynomials To subtract one integer from another, add the first integer and the additive inverse or oppo- site of the second integer. For example, 3 - 5 is evaluated as follows.

3 - 5 = 3 + ( - 5) Add the opposite. = - 2 Simplify.

Similarly, to subtract one polynomial from another, add the first polynomial and the opposite of the second polynomial. To find the opposite of a polynomial, simply negate each term. Table 5.3 lists some polynomials and their opposites.

READING CHECK

• How do you subtract one polynomial from another?

CRITICAL THINKING

What is the result when a polynomial and its opposite are added?

TABLE 5.3 Opposites of Polynomials

Polynomial Opposite

2 x - 4 - 2 x + 4

- x 2 - 2 x + 9 x 2 + 2 x - 9

6 x 3 - 12 - 6 x 3 + 12

- 3x 4 - 2 x 2 - 8 x + 3 3x 4 + 2 x 2 + 8 x - 3

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EXAMPLE 6 Subtracting polynomials

Simplify each expression. (a) (3x - 4) - (5x + 1) (b) (5x 2 + 2 x - 3) - (6 x 2 - 7x + 9) (c) (6 x 3 + x 2) - ( - 3x 3 - 9)

Solution (a) To subtract (5x + 1) from (3x - 4), we add the opposite of (5x + 1), or ( - 5x - 1).

(3x - 4) - (5x + 1) = (3x - 4) + (�5x - 1) = (3 � 5)x + (�4 � 1) = �2 x � 5

(b) The opposite of (6 x 2 - 7x + 9) is ( - 6 x 2 + 7x - 9).

(5x 2 + 2 x - 3) - (6 x 2 - 7x + 9) = (5x 2 + 2 x - 3) + ( - 6 x 2 + 7x - 9) = (5 - 6)x 2 + (2 + 7)x + ( - 3 - 9) = - x 2 + 9x - 12

(6 x 3 + x 2) - ( - 3x 3 - 9) = (6 x 3 + x 2) + (3x 3 + 9) = (6 + 3)x 3 + x 2 + 9 = 9x 3 + x 2 + 9

Now Try Exercises 57, 59, 61

NOTE: Some students prefer to subtract one polynomial from another by noting that a sub- traction sign in front of parentheses changes the signs of all of the terms within the paren- theses. For example, part (a) of the previous example could be worked as follows.

(3x - 4) - (5x + 1) = 3x - 4 - 5x - 1 = (3 - 5)x + ( - 4 - 1) = - 2 x - 5

EXAMPLE 7 Subtracting polynomials vertically

Simplify (5x 2 - 2 x + 7) - ( - 3x 2 + 3).

Solution To subtract one polynomial from another vertically, simply add the first polynomial and the opposite of the second polynomial. No x-term occurs in the second polynomial, so insert 0x.

5x 2 - 2 x + 7 3x 2 + 0x - 3 The opposite of - 3x 2 + 3 is 3x 2 - 3 or 3x 2 + 0x - 3. 8 x 2 - 2 x + 4 Add like terms in each column.

Now Try Exercise 69

Evaluating Polynomial Expressions Frequently, monomials and polynomials represent formulas that may be evaluated. We illustrate such applications in the next two examples.

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EXAMPLE 8 Writing and evaluating a monomial

Write the monomial that represents the volume of the box having a square bottom, as shown in Figure 5.5. Find the volume of the box if x = 3 feet and y = 2 feet.

Solution The volume of a box is found by multiplying the length, width, and height together. Because the length and width are both x and the height is y, the monomial xxy represents the volume of the box. This can be written x 2y. To calculate the volume, let x = 3 and y = 2 in the monomial x 2y.

x 2y = 32 # 2 = 9 # 2 = 18 cubic feet Now Try Exercise 73

Figure 5.5

EXAMPLE 9 Modeling sales of personal computers

Worldwide sales of personal computers have increased dramatically in recent years, as illustrated in Figure 5.6. The polynomial

0.7868 x 2 + 16.72 x + 122.58

approximates the number of computers sold in millions, where x = 0 corresponds to 2000, x = 1 to 2001, and so on. Estimate the number of personal computers sold in 2008 by using both the graph and the polynomial. (Source: International Data Corporation.)

Solution From the graph shown in Figure 5.7, it appears that personal computer sales were slightly more than 300 million, or about 310 million, in 2008.

Figure 5.6

4 6 8 10

100

200

300

400

Year (0 ↔ 2000)

C om

pu te

r S

al es

( m

il li

on s)

Worldwide Computer Sales

Figure 5.7

4 6 8 10

100

200

300

400

Year (0 ↔ 2000)

C om

pu te

r S

al es

( m

il li

on s)

Worldwide Computer Sales

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5.2 Putting It All Together

Monomial A number, variable, or product of numbers and variables raised to natu- ral number powers

The degree is the sum of the exponents.

The coefficient is the number in a monomial.

4 x 2y Degree: 3; coefficient: 4

- 6 x 2 Degree: 2; coefficient: - 6 - a4 Degree: 4; coefficient: - 1

x Degree: 1; coefficient: 1

- 8 Degree: 0; coefficient: - 8

Polynomial A monomial or the sum of two or more monomials

4 x 2 + 8 xy2 + 3y2 Trinomial - 9x 4 + 100 Binomial - 3x 2y3 Monomial

Like Terms Monomials containing the same vari- ables raised to the same powers

10x and - 2 x, 4 x 2 and 3x 2

5ab 2 and - ab 2, 5z and 12 z

CONCEPT EXPLANATION EXAMPLES

Addition of Polynomials To add polynomials, combine like terms.

(x 2 + 3x + 1) + (2 x 2 - 2 x + 7) = (1 + 2)x 2 + (3 - 2)x + (1 + 7) = 3x 2 + x + 8

3xy + 5xy = (3 + 5)xy = 8 xy

Opposite of a Polynomial To obtain the opposite of a polyno- mial, negate each term.

Polynomial Opposite

- 2 x 2 + x - 6 2 x 2 - x + 6 a2 - b 2 - a2 + b 2

- 3x - 18 3x + 18

Subtraction of Polynomials

To subtract one polynomial from another, add the first polynomial and the opposite of the second polynomial.

(x 2 + 3x) - (2 x 2 - 5x) = (x 2 + 3x) + ( - 2 x 2 + 5x) = (1 - 2)x 2 + (3 + 5)x = - x 2 + 8 x

Evaluating a Polynomial To evaluate a polynomial in x, sub- stitute a value for x in the expression and simplify.

To evaluate the polynomial

3x 2 - 2 x + 1 for x = 2,

substitute 2 for x and simplify.

3(2)2 - 2(2) + 1 = 9

The year 2008 corresponds to x = 8 in the given polynomial, so substitute 8 for x and evaluate the resulting expression.

0.7868 x 2 + 16.72 x + 122.58 = 0.7868(8)2 + 16.72(8) + 122.58 � 307 million

The graph and the polynomial give similiar results.

Now Try Exercise 71

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5.2 Exercises

CONCEPTS AND VOCABULARY

1. A(n) _____ is a number, a variable, or a product of numbers and variables raised to a natural number power.

2. A(n) _____ is a monomial or a sum of monomials.

3. The _____ of a monomial is the sum of the exponents of the variables.

4. The _____ of a polynomial is the degree of the term with the greatest degree.

5. A polynomial with two terms is called a(n) _____. 6. A polynomial with three terms is called a(n) _____.

7. Two monomials with the same variables raised to the same powers are terms.

8. To add two polynomials, combine terms.

9. To subtract two polynomials, add the first polynomial to the _____ of the second polynomial.

10. Polynomials can be added horizontally or _____.

PROPERTIES OF POLYNOMIALS

Exercises 11–18: Identify the degree and coefficient of the monomial.

11. 3x 2 12. y

13. - ab 14. - 2 xy

15. - 5rt 16. 8 x 2y5

17. 6 18. - 12

Exercises 19–28: Determine whether the expression is a polynomial. If it is, state how many terms and variables the polynomial contains. Then state its degree.

19. - x 20. 7z

21. 4 x 2 - 5x + 9 22. x 3 - 9

23. x + 1 x

24. 5

xy + 1

25. 3x -2y -3 26. 52a3b 4

27. - 23a4bc 28. - 7y -1z-3

Exercises 29–36: State whether the given pair of expres- sions are like terms. If they are like terms, add them.

29. 5x, - 4 x 30. x 2, 8 x 2

31. x 3, - 6 x 3 32. 4 xy, - 9xy

33. 9x, - xy 34. 5x 2y, - 3xy2

35. ab, ba 36. rt 2, - 2t 2r

ADDITION OF POLYNOMIALS

Exercises 37– 46: Add the polynomials.

37. (3x + 5) + ( - 4 x + 4)

38. ( - x + 5) + (2 x - 5)

39. (3x 2 + 4 x + 1) + (x 2 + 4 x)

40. ( - x 2 - x) + (2 x 2 + 3x - 1)

41. ( y3 + 3y2 - 5) + (3y3 + 4y - 4)

42. (4z4 + z2 - 10) + ( - z4 + 4z - 5)

43. ( - xy + 5) + (5xy - 4)

44. (2a2 + b 2) + (3a2 - 5b 2)

45. (a3b 2 + a2b 3) + (a2b 3 - a3b 2)

46. (a2 + ab + b 2) + (a2 - ab + b 2)

Exercises 47–50: Add the polynomials vertically.

47. 4 x 2 - 2 x + 1 5x 2 + 3x - 7

48. 8 x 2 + 3x + 5 - x 2 - 3x - 9

49. - x 2 + x 2 x 2 - 8 x - 1

50. a3 - 3a2b + 3ab 2 - b 3 a3 + 3a2b + 3ab 2 + b 3

SUBTRACTION OF POLYNOMIALS

Exercises 51–56: Write the opposite of the polynomial.

51. 5x 2 52. 17x + 12

53. 3a2 - a + 4

54. - b 3 + 3b

55. - 2t 2 - 3t + 4 56. 7t 2 + t - 10

Exercises 57– 66: Subtract the polynomials.

57. (3x + 1) - ( - x + 3)

58. ( - 2 x + 5) - (x + 7)

59. ( - x 2 + 6 x) - (2 x 2 + x - 2)

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60. (2y2 + 3y - 2) - ( y2 - y)

61. (z3 - 2z2 - z) - (4z2 + 5z + 1)

62. (3z4 - z) - ( - z4 + 4z2 - 5)

63. (4 xy + x 2y2) - (xy - x 2y2)

64. (a2 + b 2) - ( - a2 + b 2)

65. (ab 2) - (ab 2 + a3b)

66. (x 2 + 3xy + 4y2) - (x 2 - xy + 4y2)

Exercises 67–70: Subtract the polynomials vertically.

67. (x 2 + 2 x - 3) - (2 x 2 + 7x + 1)

68. (5x 2 - 9x - 1) - (x 2 - x + 3)

69. (3x 3 - 2 x) - (5x 3 + 4 x + 2)

70. (a2 + 3ab + 2b 2) - (a2 - 3ab + 2b 2)

APPLICATIONS

71. Exercise and Heart Rate The polynomial given by 1.6t 2 - 28t + 200 calculates the heart rate shown in Figure 5.1(b) in A Look Into Math for this section, where t represents the elapsed time in minutes since exercise stopped. (a) What is the heart rate when the athlete first stops

exercising? (b) What is the heart rate after 5 minutes? (c) Describe what happens to the heart rate after

exercise stops.

72. Cellular Phone Subscribers In the early years of cel- lular phone technology—from 1986 through 1991— the number of subscribers in millions could be mod- eled by the polynomial 0.163x 2 - 0.146 x + 0.205, where x = 1 corresponds to 1986, x = 2 to 1987, and so on. The graph illustrates this growth.

1987 0

1989 1991

Year

C el

l P

ho ne

S ub

sc ri

be rs

(m il

li on

(a) Use the graph to estimate the number of cellular phone subscribers in 1990.

(b) Use the polynomial to estimate the number of cel- lular phone subscribers in 1990.

73. Areas of Squares Write a monomial that equals the sum of the areas of the squares. Then calculate this sum for z = 10 inches.

74. Areas of Rectangles Find a monomial that equals the sum of the areas of the three rectangles. Find this sum for a = 5 yards and b = 3 yards.

75. Area of a Figure Find a polynomial that equals the area of the figure. Calculate its area for x = 6 feet.

76. Area of a Rectangle Write a polynomial that gives the area of the rectangle. Calculate its area for x = 3 feet.

77. Areas of Circles Write a polynomial that gives the sum of the areas of two circles, one with radius x and the other with radius y. Find this sum for x = 2 feet and y = 3 feet. Leave your answer in terms of p.

78. Squares and Circles Write a polynomial that gives

the sum of the areas of a square having sides of length x and a circle having diameter x. Approximate this sum to the nearest hundredth of a square foot for x = 6 feet.

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79. World Population The table lists actual and projected world population P in billions for selected years t.

(a) Find the slope of each line segment connecting consecutive data points in the table. Can these data be modeled with a line? Explain.

(b) Does the polynomial 0.077t - 148 give good estimates for the world population in year t? Explain how you decided.

Source: U.S. Census Bureau.

t 1974 1987 1999 2012

P 4 5 6 7

Source: U.S. Postal Service.

t 1963 1975 1987 2002 2007 2011

P 5¢ 13¢ 25¢ 37. 41¢ 44¢

(a) Does the polynomial 0.835t - 1635 model the data in the table exactly?

(b) Does it give approximations that are within 1.5¢ of the actual values?

WRITING ABOUT MATHEMATICS

81. Explain what the terms monomial, binomial, trino- mial, and polynomial mean. Give an example of each.

82. Explain how to determine the degree of a polynomial having one variable. Give an example.

83. Explain how to obtain the opposite of a polynomial. Give an example.

84. Explain how to subtract two polynomials. Give an example.

80. Price of a Stamp The table lists the price P of a first- class postage stamp for selected years t.

Checking Basic ConceptsSECTIONS5.1 and 5.2 1. Evaluate each expression. (a) - 52 (b) 32 - 23

2. Simplify each expression. (a) 103 # 105 (b) (3x 2) ( - 4 x 5) (c) (a3b)2 (d) a x

z3 b4

3. Simplify each expression. (a) (4y3)0 (b) (x 3)2(3x 4)2 (c) 2a2(5a3 - 7)

4. State the number of terms and variables in the polynomial 5x 3y - 2 x 2y + 5. What is its degree?

5. A box has a rectangular bottom twice as long as it is wide.

(a) If the bottom has width w and the box has height h, write a monomial that gives the vol- ume of the box.

(b) Find the volume of the box for w = 12 inches and h = 10 inches.

6. Simplify each expression. (a) (2a2 + 3a - 1) + (a2 - 3a + 7) (b) (4z 3 + 5z) - (2z 3 - 2z + 8) (c) (x 2 + 2 xy + y2) - (x 2 - 2 xy + y2)

5.3 Multiplication of Polynomials Multiplying Monomials ● Review of the Distributive Properties ● Multiplying Monomials and Polynomials ● Multiplying Polynomials

A LOOK INTO MATH N The study of polynomials dates back to Babylonian civilization in about 1800–1600 B.C. Many eighteenth-century mathematicians devoted their entire careers to the study of poly- nomials. Today, polynomials still play an important role in mathematics, often being used to approximate unknown quantities. In this section we discuss the basics of multiplying polynomials. (Source: Historical Topics for the Mathematics Classroom, Thirty-first Yearbook, NCTM.)

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Multiplying Monomials A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers. To multiply monomials, we often use the product rule for exponents.

EXAMPLE 1 Multiplying monomials

Multiply. (a) - 5x 2 # 4 x 3 (b) (7xy4)(x 3y2) Solution (a) - 5x 2 # 4 x 3 = ( - 5)(4)x 2x 3 Commutative property = - 20x 2 + 3 The product rule = - 20x 5 Simplify.

(b) (7xy4)(x 3y2) = 7xx 3y4y2 Commutative property = 7x 1 + 3y4 + 2 The product rule = 7x 4y6 Simplify.

Now Try Exercises 9, 13

READING CHECK

• Which rule for exponents is commonly used to multiply monomials?

Review of the Distributive Properties Distributive properties are used frequently for multiplying monomials and polynomials. For all real numbers a, b, and c,

a(b + c) = ab + ac and a(b - c) = ab - ac.

The first distributive property above can be visualized geometrically. For example,

3(x + 2) = 3x + 6

is illustrated in Figure 5.8. The dimensions of the large rectangle are 3 by x + 2, and its area is 3(x + 2). The areas of the two small rectangles, 3x and 6, equal the area of the large rectangle. Therefore 3(x + 2) = 3x + 6.

In the next example we use the distributive properties to multiply expressions.

EXAMPLE 2 Using distributive properties

Multiply. (a) 2(3x + 4) (b) (3x 2 + 4)5 (c) - x(3x - 6)

Solution

(a) 2(3x + 4) = 2 # 3x + 2 # 4 = 6 x + 8 (b) (3x 2 + 4)5 = 3x 2 # 5 + 4 # 5 = 15x 2 + 20 (c) - x(3x - 6) = - x # 3x + x # 6 = - 3x 2 + 6 x Now Try Exercises 15, 19, 21

Figure 5.8 Area: 3x + 6

x + 2

3 3x 6

x 2

Distributive Property 3(x � 2) � 3x � 6

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Multiplying Monomials and Polynomials A monomial consists of one term, whereas a polynomial consists of one or more terms separated by + or - signs. To multiply a monomial by a polynomial, we apply the distribu- tive properties and the product rule.

READING CHECK

• What properties are com- monly used to multiply a monomial and a polynomial?

EXAMPLE 3 Multiplying monomials and polynomials

Multiply. (a) 9x(2 x 2 - 3) (b) (5x - 8)x 2 (c) - 7(2 x 2 - 4 x + 6) (d) 4 x 3(x 4 + 9x 2 - 8)

Solution

(a) 9x(2 x 2 - 3) = 9x # 2 x 2 - 9x # 3 Distributive property = 18 x 3 - 27x The product rule

(b) (5x - 8)x 2 = 5x # x 2 - 8 # x 2 Distributive property = 5x 3 - 8 x 2 The product rule

(d) 4 x 3(x 4 + 9x 2 - 8) = 4 x 3 # x 4 + 4 x 3 # 9x 2 - 4 x 3 # 8 Distributive property = 4 x 7 + 36 x 5 - 32 x 3 The product rule

Now Try Exercises 23, 25, 27, 29

We can also multiply monomials and polynomials that contain more than one variable.

EXAMPLE 4 Multiplying monomials and polynomials

Multiply. (a) 2 xy(7x 2y3 - 1) (b) - ab(a2 - b 2)

Solution

(a) 2 xy(7x 2y3 - 1) = 2 xy # 7x 2y3 - 2 xy # 1 Distributive property = 14 xx 2yy3 - 2 xy Commutative property = 14 x 3y4 - 2 xy The product rule

(b) - ab(a2 - b 2) = - ab # a2 + ab # b 2 Distributive property = - aa2b + abb 2 Commutative property = - a3b + ab 3 The product rule

Now Try Exercises 31, 35

Multiplying Polynomials Monomials, binomials, and trinomials are examples of polynomials. Recall that a monomial has one term, a binomial has two terms, and a trinomial has three terms. In the next example we multiply two binomials, using both geometric and symbolic techniques.

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EXAMPLE 5 Multiplying binomials

Multiply (x + 4)(x + 2) (a) geometrically and (b) symbolically.

Solution (a) To multiply (x + 4)(x + 2) geometrically, draw a rectangle x + 4 long and x + 2

wide, as shown in Figure 5.9(a). The area of this rectangle equals length times width, or (x + 4)(x + 2). The large rectangle can be divided into four smaller rectangles, which have areas of x 2, 4 x, 2 x, and 8, as shown in Figure 5.9(b). Thus

(x + 4)(x + 2) = x 2 � 4 x � 2 x + 8 = x 2 � 6 x + 8.

(b) To multiply (x + 4)(x + 2) symbolically, apply the distributive property two times.

(x � 4) (x + 2) = (x � 4) (x) + (x � 4) (2) = x # x + 4 # x + x # 2 + 4 # 2 = x 2 � 4 x � 2 x + 8 = x 2 � 6 x + 8

r Now Try Exercise 39

The distributive properties used in part (b) of the previous example show that if we want to multiply (x + 4) by (x + 2), we should multiply every term in x + 4 by every term in x + 2.

(x + 4)(x + 2) = x 2 + 2 x + 4 x + 8 = x 2 + 6 x + 8

NOTE: This process of multiplying binomials is sometimes called FOIL. This acro- nym may be used to remind us to multiply the first terms (F ), outside terms (O), inside terms (I ), and last terms (L). The FOIL process is a shortcut for the process used in Example 5(b).

Multiply the First terms to obtain x 2. (x + 4)(x + 2)

Multiply the Outside terms to obtain 2 x. (x + 4)(x + 2)

Multiply the Inside terms to obtain 4 x. (x + 4)(x + 2)

Multiply the Last terms to obtain 8. (x + 4)(x + 2)

MULTIPLYING POLYNOMIALS

The product of two polynomials may be found by multiplying every term in the first polynomial by every term in the second polynomial and then combining like terms.

READING CHECK

• What kind of polynomials can be multiplied using the FOIL method?

Figure 5.9

x + 2

x + 4

(a) Area = (x + 4)(x + 2)

2 2x

x 4

4xx

(b) Area = x2 + 4x + 2x + 8

The following statement summarizes how to multiply two polynomials in general.

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EXAMPLE 6 Multiplying binomials

Multiply. Draw arrows to show how each term is found. (a) (3x + 2)(x + 1) (b) (1 - x)(1 + 2 x) (c) (4 x - 3)(x 2 - 2 x)

Solution

(a) (3x + 2)(x + 1) = 3x # x + 3x # 1 + 2 # x + 2 # 1 = 3x 2 � 3x � 2 x + 2 = 3x 2 � 5x + 2

(b) (1 - x)(1 + 2 x) = 1 # 1 + 1 # 2 x - x # 1 - x # 2 x = 1 � 2 x � x - 2 x 2

= 1 � x - 2 x 2

Now Try Exercises 51, 53, 59

The FOIL process may be helpful for remembering how to multiply two binomials, but it cannot be used for every product of polynomials. In the next example, the general process for multiplying polynomials is used to find products of binomials and trinomials.

EXAMPLE 7 Multiplying polynomials

Multiply. (a) (2 x + 3)(x 2 + x - 1) (b) (a - b)(a2 + ab + b 2) (c) (x 4 + 2 x 2 - 5)(x 2 + 1)

Solution (a) Multiply every term in (2 x + 3) by every term in (x 2 + x - 1).

(2 x + 3)(x 2 + x - 1) = 2 x # x 2 + 2 x # x - 2 x # 1 + 3 # x 2 + 3 # x - 3 # 1 = 2 x 3 � 2 x 2 � 2 x � 3x 2 � 3x - 3 = 2 x 3 � 5x 2 � x - 3

(b) (a - b)(a2 + ab + b 2) = a # a2 + a # ab + a # b 2 - b # a2 - b # ab - b # b 2 = a3 � a2b � ab 2 � a2b � ab 2 - b 3

= a3 - b 3

(c) (x 4 + 2 x 2 - 5)(x 2 + 1) = x 4 # x 2 + x 4 # 1 + 2 x 2 # x 2 + 2 x 2 # 1 - 5 # x 2 - 5 # 1 = x 6 � x 4 � 2 x 4 � 2 x 2 � 5x 2 - 5 = x 6 � 3x 4 � 3x 2 - 5

Now Try Exercises 63, 67, 69

STUDY TIP

Even if you know exactly how to do a math problem correctly, a simple computational error will often cause you to get an incorrect answer. Be sure to take your time on simple calculations.

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Polynomials can be multiplied vertically in a manner similar to multiplication of real numbers. For example, multiplication of 123 times 12 is performed as follows.

1 2 3

* 1 2 2 4 6

1 2 3

1 4 7 6

A similar method can be used to multiply polynomials vertically.

EXAMPLE 8 Multiplying polynomials vertically

Multiply 2 x 2 - 4 x + 1 and x + 3 vertically.

Solution Write the polynomials vertically. Then multiply every term in the first polynomial by each term in the second polynomial. Arrange the results so that like terms are in the same column.

2 x 2 - 4 x + 1 x + 3

6 x 2 - 12 x + 3 2 x 3 - 4 x 2 + x 2 x 3 + 2 x 2 - 11x + 3

Multiply top polynomial by 3. Multiply top polynomial by x.

Add each column.

Now Try Exercise 71

MAKING CONNECTIONS

Vertical and Horizontal Formats

Whether you decide to add, subtract, or multiply polynomials vertically or horizontally, remember that the same answer is obtained either way.

EXAMPLE 9 Finding the volume of a box

A box has a width 3 inches less than its height and a length 4 inches more than its height. (a) If h represents the height of the box, write a polynomial that represents the volume of

the box. (b) Use this polynomial to calculate the volume of the box if h = 10 inches.

Solution (a) If h is the height, then h - 3 is the width and h + 4 is the length, as illustrated in

Figure 5.10. Its volume equals the product of these three expressions.

h(h - 3)(h + 4) = (h 2 - 3h)(h + 4)

= h 2 # h + h 2 # 4 - 3h # h - 3h # 4 = h 3 + 4h 2 - 3h 2 - 12h = h 3 + h 2 - 12h

(b) If h = 10, then the volume is

103 + 102 - 12(10) = 1000 + 100 - 120 = 980 cubic inches.

Now Try Exercise 79

Figure 5.10

h – 3

h + 4

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5.3 Putting It All Together

Distributive Properties For all real numbers a, b, and c,

a(b + c) = ab + ac and a(b - c) = ab - ac.

5(x + 3) = 5x + 15, 3(x - 6) = 3x - 18, and - 2 x(3 - 5x 3) = - 6 x + 10x 4

Multiplying Polynomials The product of two polynomials may be found by multiplying every term in the first polynomial by every term in the second polynomial and then com- bining like terms.

3x(5x 2 + 2 x - 7) = 3x # 5x 2 + 3x # 2 x - 3x # 7 = 15x 3 + 6 x 2 - 21x

(x + 2)(7x - 3) = x # 7x - x # 3 + 2 # 7x - 2 # 3 = 7x 2 - 3x + 14 x - 6 = 7x 2 + 11x - 6

CONCEPT EXPLANATION EXAMPLES

5.3 Exercises

CONCEPTS AND VOCABULARY

1. The equation x 2 # x 3 = x 5 illustrates what rule of exponents?

2. The equation 3(x - 2) = 3x - 6 illustrates what property?

3. The product of two polynomials may be found by multiplying every in the first polynomial by every in the second polynomial and then com- bining like terms.

4. Polynomials can be multiplied horizontally or _____.

MULTIPLICATION OF MONOMIALS

Exercises 5–14: Multiply.

5. x 2 # x 5 6. - a # a5 7. - 3a # 4a 8. 7x # 5x 9. 4 x 3 # 5x 2 10. 6b 6 # 3b 5 11. xy2 # 4 xy 12. 3ab # ab 2 13. ( - 3xy2)(4 x 2y) 14. ( - r 2t 2)( - r 3t)

MULTIPLICATION OF MONOMIALS AND POLYNOMIALS

Exercises 15–36: Multiply and simplify the expression.

15. 3(x + 4) 16. - 7(4 x - 1)

19. (4 - z)z 20. 3z(1 - 5z)

17. - 5(9x + 1)

18. 10(1 - 6 x)

21. - y(5 + 3y)

22. (2y - 8)2y

23. 3x(5x 2 - 4) 24. - 6 x(2 x 3 + 1)

27. - 8(4t 2 + t + 1)

28. 7(3t 2 - 2t - 5)

31. xy(x + y)

32. ab(2a - 3b)

25. (6 x - 6)x 2

26. (1 - 2 x 2)3x 2

29. n2( - 5n2 + n - 2)

30. 6n3(2 - 4n + n2)

33. x 2(x 2y - xy2)

34. 2y2(xy - 5)

35. - ab(a3 - 2b 3)

36. 5rt(r 2 + 2rt + t 2)

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MULTIPLICATION OF POLYNOMIALS

Exercises 37–42: (Refer to Example 5.) Multiply the given expression (a) geometrically and (b) symbolically.

37. x(x + 3)

38. 2 x(x + 5)

39. (x + 2)(x + 2)

40. (x + 1)(x + 3)

41. (x + 3)(x + 6)

42. (x + 5)(x + 2)

43. (x + 3)(x + 5)

44. (x - 4)(x - 7)

45. (x - 8)(x - 9)

46. (x + 10)(x + 10)

51. (10y + 7)( y - 1) 52. ( y + 6)(2y + 7)

57. (x - 1)(x 2 + 1)

58. (x + 2)(x 2 - x)

47. (3z - 2)(2z - 5) 48. (z + 6)(2z - 1)

53. (5 - 3a)(1 - 2a) 54. (4 - a)(5 + 3a)

59. (x 2 + 4)(4 x - 3)

60. (3x 2 - 1)(3x 2 + 1)

49. (8b - 1)(8b + 1) 50. (3t + 2)(3t - 2)

55. (1 - 3x)(1 + 3x) 56. (10 - x)(5 - 2 x)

61. (2n + 1)(n2 + 3) 62. (2 - n2)(1 + n2)

Exercises 43–70: Multiply and simplify the expression.

63. (m + 1)(m2 + 3m + 1)

64. (m - 2)(m2 - m + 5)

65. (3x - 2)(2 x 2 - x + 4)

66. (5x + 4)(x 2 - 3x + 2)

67. (x + 1)(x 2 - x + 1)

68. (x - 2)(x 2 + 4 x + 4)

69. (4b 2 + 3b + 7)(b 2 + 3)

70. ( - 3a2 - 2a + 1)(3a2 - 3)

Exercises 71–76: Multiply the polynomials vertically.

71. (x + 2)(x 2 - 3x + 1)

72. (2y - 3)(3y2 - 2y - 2)

73. (a - 2)(a2 + 2a + 4)

74. (b - 3)(b 2 + 3b + 9)

75. (3x 2 - x + 1)(2 x 2 + 1)

76. (2 x 2 - 3x - 5)(2 x 2 + 3)

77. Thinking Generally If a polynomial with m terms and a polynomial with n terms are multiplied, how many terms are there in the product before like terms are combined?

78. Thinking Generally When a polynomial with m terms is multiplied by a second polynomial, the product contains k terms before like terms are combined. How many terms does the second polynomial contain?

APPLICATIONS

79. Volume of a Box (Refer to Example 9.) A box has a width 4 inches less than its height and a length 2 inches more than its height. (a) If h is the height of the box, write a polynomial

that represents the volume of the box. (b) Use this polynomial to calculate the volume for

h = 25 inches.

80. Surface Area of a Box Use the drawing of the box to write a polynomial that represents each of the following.

b + 1

2b + 1

(a) The area of its bottom (b) The area of its front (c) The area of its right side (d) The total area of its six sides

81. Perimeter of a Pen A rectangular pen for a pet has a perimeter of 100 feet. If one side of the pen has length x, then its area is given by x(50 - x). (a) Multiply this expression. (b) Evaluate the expression obtained in part (a) for

x = 25.

82. Rectangular Garden A rectangular garden has a perimeter of 500 feet. (a) If one side of the garden has length x, then write

a polynomial expression that gives its area. Mul- tiply this expression completely.

(b) Evaluate the expression for x = 50 and interpret your answer.

83. Surface Area of a Cube Write a polynomial that rep- resents the total area of the six sides of the cube hav- ing edges with length x + 1.

x + 1

x + 1 x + 1

84. Surface Area of a Sphere The surface area of a sphere with radius r is 4pr 2. Write a polynomial that gives the surface area of a sphere with radius x + 2. Leave your answer in terms of p.

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85. Toy Rocket A toy rocket is shot straight up into the air. Its height h in feet above the ground after t sec- onds is represented by the expression t(64 - 16t). (a) Multiply this expression. (b) Evaluate the expression obtained in part (a) and

the given expression for t = 2. (c) Are your answers in part (b) the same? Should

they be the same?

86. Toy Rocket on the Moon (Refer to the preceding exercise.) If the same toy rocket were flown on the moon, then its height h in feet after t seconds would be t164 - 52 t2. (a) Multiply this expression. (b) Evaluate the expression obtained in part (a) and

the given expression for t = 2. Did the rocket go higher on the moon?

WRITING ABOUT MATHEMATICS

87. Explain how the acronym FOIL relates to multiplying two binomials, such as x + 3 and 2 x + 1.

88. Does the FOIL method work for multiplying a bino- mial and a trinomial? Explain.

89. Explain in words how to multiply any two polynomi- als. Give an example.

90. Give two properties of real numbers that are used for multiplying 3x(5x 2 - 3x + 2). Explain your answer.

Working with Real DataGroup Activity

Biology Some types of worms have a remarkable ability to live without moisture. The following table from one study shows the number of worms W surviv- ing after x days without moisture.

(a) Use the equation W = - 0.0014 x 2 - 0.076 x + 50 to find W for each x-value in the table.

Directions: Form a group of 2 to 4 people. Select someone to record the group’s responses for this activity. All members of the group should work cooperatively to answer the questions. If your instructor asks for your results, each member of the group should be prepared to respond.

(b) Discuss how well this equation approximates the data.

Source: D. Brown and P. Rothery, Models in Biology.

x (days) 0 20 40 80 120 160

W (worms) 50 48 45 36 20 3

5.4 Special Products Product of a Sum and Difference ● Squaring Binomials ● Cubing Binomials

A LOOK INTO MATH N Polynomials are often used to approximate real-world phenomena in applications. Poly- nomials have played an important role in the development of everyday products such as tablet computers, cell phones, and automobiles. Even digital images in computers and interest calculations at a bank make use of polynomials. In this section we discuss how to multiply some special types of binomials.

Product of a Sum and Difference Products of the form (a + b)(a - b) occur frequently in mathematics. Other examples include

(x + y)(x - y) and (2r + 3t)(2r - 3t).

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These products can always be multiplied by using the techniques discussed in Section 5.3. However, there is a faster way to multiply these special products.

(a + b)(a - b) = a # a � a # b � b # a - b # b = a2 � ab � ba - b 2

= a2 - b 2

In words, the product of a sum of two numbers and their difference equals the difference of their squares. We generalize this method as follows.

STUDY TIP

In mathematics, there are often several correct ways to perform a particular process. If your instructor does not require you to use a specified method, choose the one that works best for you.

READING CHECK

• Explain in words how you can find the product of the sum of two numbers and their difference.

PRODUCT OF A SUM AND DIFFERENCE

For any real numbers a and b,

(a + b)(a - b) = a2 - b 2.

EXAMPLE 1 Finding products of sums and differences

Multiply. (a) (x + y)(x - y) (b) (z - 2)(z + 2) (c) (2r + 3t)(2r - 3t) (d) (5m2 - 4n2)(5m2 + 4n2)

Solution (a) If we let a = x and b = y, then we can apply the rule

(a + b)(a - b) = a2 - b 2.

Thus

(x + y)(x - y) = (x)2 - ( y)2

= x 2 - y2.

(b) Because the expressions (z + 2)(z - 2) and (z - 2)(z + 2) are equal by the commu- tative property, we can apply the formula for the product of a sum and difference.

(z - 2)(z + 2) = (z)2 - (2)2

= z2 - 4

(2r + 3t)(2r - 3t) = (2r)2 - (3t)2

= 4r 2 - 9t 2

(d) (5m2 - 4n2)(5m2 + 4n2) = (5m2)2 - (4n2)2 = 25m4 - 16n4

Now Try Exercises 7, 13, 17

The next example demonstrates how the product of a sum and difference can be used to find some products of numbers mentally.

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EXAMPLE 2 Finding a product

Use the product of a sum and difference to find 22 # 18. Solution Because 22 = 20 + 2 and 18 = 20 - 2, rewrite and evaluate 22 # 18 as follows. 22 # 18 = (20 + 2)(20 - 2) Rewrite 22 as 20 + 2 and 18 as 20 - 2. = 202 - 22 Product of a sum and difference = 400 - 4 Evaluate exponents. = 396 Subtract.

Now Try Exercise 21

Squaring Binomials Because each side of the square shown in Figure 5.11 has length (a + b), its area equals

(a + b)(a + b),

which can be written as (a + b)2. We can multiply this expression as follows.

(a + b)2 = (a + b)(a + b) = a2 � ab � ba + b 2

= a2 � 2ab + b 2

This result is illustrated geometrically in Figure 5.12, where the area of the large square is (a + b)2. This area can also be found by adding the areas of the four small rectangles.

a2 � ab � ba + b 2 = a2 � 2ab + b 2

The geometric and symbolic results are the same. Note that to obtain the middle term, 2ab, we can multiply the two terms, a and b, in the binomial and double the result.

A similar product that is also the square of a binomial can be calculated as

(a - b)2 = (a - b)(a - b) = a2 � ab � ba + b 2

= a2 � 2ab + b 2.

These results are summarized as follows.

READING CHECK

• Explain in words how you can square a binomial.

SQUARING A BINOMIAL

For any real numbers a and b,

(a + b)2 = a2 + 2ab + b 2 and (a - b)2 = a2 - 2ab + b 2.

That is, the square of a binomial equals the square of the first term, plus (or minus) twice the product of the two terms, plus the square of the last term.

NOTE: (a + b)2 3 a2 + b 2. Do not forget the middle term when squaring a binomial.

EXAMPLE 3 Squaring a binomial

Multiply. (a) (x + 3)2 (b) (2 x - 5)2 (c) (1 - 5y)2 (d) (7a2 + 3b)2

Figure 5.11

a + b

Area � (a � b)2

Figure 5.12

a b

(a � b)2 � a2 � 2ab � b 2

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Solution (a) Let a = x and b = 3 in the formula (a + b)2 = a2 + 2ab + b 2.

(x + 3)2 = (x)2 + 2(x)(3) + (3)2

= x 2 + 6 x + 9

(b) Apply the formula (a - b)2 = a2 - 2ab + b 2 with a = 2 x and b = 5.

(2 x - 5)2 = (2 x)2 - 2(2 x)(5) + (5)2

= 4 x 2 - 20x + 25

(d) (7a2 + 3b)2 = (7a2)2 + 2(7a2)(3b) + (3b)2 = 49a4 + 42a2b + 9b 2

Now Try Exercises 27, 29, 35, 39

MAKING CONNECTIONS

Multiplying Binomials and Special Products

If you forget these special products, you can still multiply polynomials by using earlier techniques. For example, the binomial in Example 3(b) can be multiplied as

(2 x - 5)2 = (2 x - 5)(2 x - 5)

= 2 x # 2 x - 2 x # 5 - 5 # 2 x + 5 # 5 = 4 x 2 - 10x - 10x + 25 = 4 x 2 - 20x + 25.

N REAL-WORLD CONNECTION NASA first developed digital pictures because they were easy to transmit through space and because they provided clear images. A digital image from outer space is shown in Figure 5.13.

Today, digital cameras are readily available, and the Internet uses digital images exclu- sively. The next example shows how polynomials relate to digital pictures.

EXAMPLE 4 Calculating the size of a digital picture

A digital picture comprises tiny square units called pixels. Shading individual pixels creates a picture. A simplified version of a digital picture of the letter T is shown in Figure 5.14. This picture includes an image of the letter T that measures 3 pixels by 3 pixels and a 1-pixel border. (a) If a square digital image measures x pixels by x pixels, then a picture that includes the

image and a 1-pixel border will measure x + 2 pixels by x + 2 pixels. Find a polyno- mial that gives the total number of pixels in the picture, including the border.

(b) Let x = 3 and evaluate the polynomial. Does it agree with Figure 5.14?

Solution (a) The total number of pixels equals (x + 2) times (x + 2), or (x + 2)2.

(x + 2)2 = x 2 + 4 x + 4

(b) For x = 3, the polynomial evaluates to 32 + 4 # 3 + 4 = 25, the total number of pixels. This result agrees with Figure 5.14, which has a total of 5 # 5 = 25 pixels with a 3 pixel by 3 pixel image of the letter T inside.

Now Try Exercise 79

Figure 5.13 Digital Picture

Figure 5.14

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Cubing Binomials To calculate the volume of the cube shown in Figure 5.15, we find the product of its length, width, and height. Because all sides have the same measure, its volume is (x + 2)3. That is, the volume equals the cube of x + 2.

To multiply the expression (x + 2)3, we proceed as follows.

(x + 2)3 = (x + 2)(x � 2)2 a3 = a # a2

= (x + 2)(x 2 � 4 x � 4) Square the binomial.

= x # x 2 + x # 4 x + x # 4 + 2 # x 2 + 2 # 4 x + 2 # 4 Multiply the polynomials. = x 3 + 4 x 2 + 4 x + 2 x 2 + 8 x + 8 Simplify terms. = x 3 + 6 x 2 + 12 x + 8 Combine like terms.

Figure 5.15 Volume = (x + 2)3

x + 2

x + 2 x + 2

EXAMPLE 5 Cubing a binomial

Multiply (2z - 3)3.

Solution

(2z - 3)3 = (2z - 3)(2z � 3)2 a3 = a # a2

= (2z - 3)(4z2 � 12z � 9) Square the binomial.

= 8z3 - 24z2 + 18z - 12z2 + 36z - 27 Multiply the polynomials. = 8z3 - 36z2 + 54z - 27 Combine like terms.

NOTE: (2z - 3)3 3 (2z)3 - (3)3 = 8z3 - 27.

Now Try Exercise 47

CRITICAL THINKING

Suppose that a student is convinced that the expressions

(x + y)3 and x3 + y 3

are equal. How could you convince the student otherwise?

EXAMPLE 6 Calculating interest

If a savings account pays x percent annual interest, where x is expressed as a decimal, then after 3 years a sum of money will grow by a factor of (1 + x)3. (a) Multiply this expression. (b) Evaluate the expression for x = 0.10 (or 10%), and interpret the result.

Solution (a) (1 + x)3 = (1 + x)(1 + x)2 a3 = a # a2 = (1 + x)(1 + 2 x + x 2) Square the binomial. = 1 + 2 x + x 2 + x + 2 x 2 + x 3 Multiply the polynomials. = 1 + 3x + 3x 2 + x 3 Combine like terms.

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(b) Let x = 0.1 in the expression 1 + 3x + 3x 2 + x 3.

1 + 3(0.1) + 3(0.1)2 + (0.1)3 = 1.331

The sum of money will increase by a factor of 1.331. For example, $1000 deposited in this account will grow to $1331 in 3 years.

Now Try Exercise 75

5.4 Putting It All Together

Product of a Sum and Difference

For any real numbers x and y,

(x + y)(x - y) = x 2 - y2. (x + 6)(x - 6) = x 2 - 36, (2 x - 3)(2 x + 3) = 4 x 2 - 9, and (x 2 + y2)(x 2 - y2) = x 4 - y4

Squaring a Binomial For all real numbers x and y,

(x + y)2 = x 2 + 2 xy + y2 and (x - y)2 = x 2 - 2 xy + y2.

(x + 4)2 = x 2 + 8 x + 16, (5x - 2)2 = 25x 2 - 20x + 4, and (x 2 + y2)2 = x 4 + 2 x 2y2 + y4

Cubing a Binomial Multiply the binomial by its square. (x + 3)3

= (x + 3)(x + 3)2

= (x + 3)(x 2 + 6 x + 9) = x 3 + 6 x 2 + 9x + 3x 2 + 18 x + 27 = x 3 + 9x 2 + 27x + 27

CONCEPT EXPLANATION EXAMPLES

5.4 Exercises

CONCEPTS AND VOCABULARY

1. (a + b)(a - b) = _____

2. (a + b)2 = _____

3. (a - b)2 = _____

4. (a + b)3 = (a + b) # _____ 5. (True or False?) The two expressions (x + y)2 and

x 2 + y2 are equal for all real numbers x and y.

6. (True or False?) The two expressions (r - t)2 and r 2 - t 2 are equal for all real numbers r and t.

PRODUCT OF A SUM AND DIFFERENCE

Exercises 7–20: Multiply.

7. (x - 3)(x + 3)

8. (x + 6)(x - 6)

9. (4 x - 1)(4 x + 1) 10. (10x + 3)(10x - 3)

11. (1 + 2a)(1 - 2a) 12. (4 - 9b)(4 + 9b)

13. (2 x + 3y)(2 x - 3y) 14. (5r - 6t)(5r + 6t)

15. (ab - 5)(ab + 5)

16. (2 xy + 7)(2 xy - 7)

17. (a2 - b 2)(a2 + b 2)

18. (3x 2 + y2)(3x 2 - y2)

19. (x 3 - y3)(x 3 + y3)

20. (2a4 + b 4)(2a4 - b 4)

Exercises 21–26: (Refer to Example 2.) Use the product of a sum and a difference to evaluate the expression.

21. 101 # 99 22. 52 # 48 23. 23 # 17 24. 29 # 31 25. 90 # 110 26. 38 # 42

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SQUARING BINOMIALS

Exercises 27– 40: Multiply.

27. (a - 2)2

28. (x - 7)2

31. (3b + 5)2 32. (7t + 10)2

35. (1 - b)2 36. (1 - 4a)2

29. (2 x + 3)2

30. (7x - 2)2

33. 134 a - 422

37. (5 + y3)2 38. (9 - 5x 2)2

34. 115 a + 122

39. (a2 + b)2

40. (x 3 - y3)2

CUBING BINOMIALS

Exercises 41– 48: Multiply.

41. (a + 1)3

42. (b + 4)3

43. (x - 2)3

44. ( y - 7)3

45. (2 x + 1)3

46. (4z + 3)3

47. (6u - 1)3 48. (5v + 3)3

MULTIPLICATION OF POLYNOMIALS

Exercises 49–66: Multiply, using any appropriate method.

(b) Find the sum of the areas of the smaller rectan- gles inside the large square.

49. 4(5x + 9) 50. (2 x + 1)(3x - 5)

51. (x - 5)(x + 7)

52. (x + 10)(x + 10)

53. (3x - 5)2 54. (x - 3)(x + 9)

55. (5x + 3)(5x + 4) 56. - x 3(x 2 - x + 1)

57. (4b - 5)(4b + 5) 58. (x + 5)3

59. - 5x(4 x 2 - 7x + 2)

60. (4 x 2 - 5)(4 x 2 + 5)

61. (4 - a)3 62. 2 x(x - 3)3

63. x(x + 3)2

64. (x - 1)2(x + 1)

65. (x + 2)(x - 2)(x + 1)(x - 1)

66. (x - y)(x + y)(x 2 + y2)

67. Thinking Generally Multiply (an + b n)(an - b n). 68. Thinking Generally Multiply (an + b n)2.

APPLICATIONS

Exercises 69–72: Do each part and verify that your answers are the same.

(a) Find the area of the large square by multiplying its length and width.

69. 2

x 2

70. 4

a 4

71.

2x 3

72. x

3y x

Exercises 73 and 74: Find a polynomial that represents the following.

(a) The outside surface area given by the six sides of the cube

(b) The volume of the cube

73.

x + 5

x + 5 x + 5

74.

2x + 1

2x + 1 2x + 1

75. Compound Interest (Refer to Example 6.) If a sum of money is deposited in a savings account that is paying x percent annual interest (expressed as a deci- mal), then this sum of money increases by a factor of (1 + x)2 after 2 years. (a) Multiply this expression. (b) Evaluate the polynomial expression found in

part (a) for an annual interest rate of 10%, or x = 0.10, and interpret the answer.

76. Compound Interest If a sum of money is deposited in a savings account that is paying x percent annual interest, then this sum of money increases by a factor of 11 + 1100x23 after 3 years. (a) Multiply this expression.

(b) Evaluate the polynomial expression in part (a) for an annual interest rate of 8%, or x = 8, and interpret the answer.

77. Probability If there is an x percent chance of rain on each of two consecutive days, then the expres- sion (1 - x)2 gives the percent chance that neither day will have rain. Assume that all percentages are expressed as decimals.

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(a) Multiply this expression. (b) Evaluate the polynomial expression in part (a)

for a 50% chance of rain, or x = 0.50, and inter- pret the answer.

78. Probability If there is an x percent chance of rolling a 6 with one die, then the expression (1 - x)3 gives the percent chance of not rolling a 6 with three dice. Assume that all percentages are expressed as deci- mals or fractions. (a) Multiply this expression. (b) Evaluate the polynomial expression found in part

(a) for a 16.6% chance of rolling a 6, or x = 16 , and interpret the answer.

79. Swimming Pool A square swimming pool has an 8-foot-wide sidewalk around it. (a) If the sides of the pool have length z, as shown in

the accompanying figure, find a polynomial that gives the area of the sidewalk.

(b) Evaluate the polynomial in part (a) for z = 60 and interpret the answer.

80. Digital Picture (Refer to Example 4.) Suppose that a digital picture, including its border, is x + 2 pixels by x + 2 pixels and that the actual image inside the border is x - 2 pixels by x - 2 pixels, as shown in the following figure.

x + 2

x + 2 x – 2

x – 2

(a) Find a polynomial that gives the number of pixels in the border.

(b) Evaluate the polynomial in part (a) for x = 5. (c) Sketch a digital picture of the letter H with

x = 5. Does the picture agree with the answer in part (b)?

(d) Digital pictures typically have large values for x. If a picture has x = 500, find the total number of pixels in its border.

WRITING ABOUT MATHEMATICS

81. Explain why (a + b)2 does not equal a2 + b 2 in general for real numbers a and b.

82. Explain how to find the cube of a binomial.

Checking Basic ConceptsSECTIONS5.3 and 5.4 1. Multiply each expression. (a) ( - 3xy4)(5x 2y) (b) - x(6 - 4 x) (c) 3ab(a2 - 2ab + b 2)

2. Multiply each expression. (a) (x + 3)(4 x - 3) (b) (x 2 - 1)(2 x 2 + 2) (c) (x + y)(x 2 - xy + y2)

3. Multiply each expression. (a) (5x + 2)(5x - 2) (b) (x + 3)2

4. Complete each part and verify that your answers are the same.

(a) Find the area of the large square by squaring the length of one of its sides.

(b) Find the sum of the areas of the smaller rect- angles inside the large square.

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NEW VOCABULARY

n Scientific notation

5.5 Integer Exponents and the Quotient Rule Negative Integers as Exponents ● The Quotient Rule ● Other Rules for Exponents ● Scientific Notation

A LOOK INTO MATH N In 2009, astronomers discovered a large planet that orbits a distant star. The planet, named WASP-17b, is about 5,880,000,000,000,000 miles from Earth. Also in 2009, the H1N1 virus was identified in a worldwide influenza pandemic. A typical flu virus measures about 0.00000468 inch across. In this section, we will discuss how integer exponents can be used to write such numbers in scientific notation. (Source: Scientific American.)

READING CHECK

• How is a negative integer power on a base related to the corresponding posi- tive integer power on that base?

TABLE 5.4 Powers of 2

Power of 2 Value

21 2

20 1

2-1 12 = 1 21

2-2 14 = 1 22

Decrease exponent by 1

Divide by 2

Table 5.4 shows that 2-1 = 1 21

and 2-2 = 1 22

. In other words, if the exponent on 2 is nega- tive, then the expression is equal to the reciprocal of the corresponding expression with a positive exponent on 2. This discussion suggests the following definition for negative inte- ger exponents.

NEGATIVE INTEGER EXPONENTS

Let a be a nonzero real number and n be a positive integer. Then

a -n = 1

an .

That is, a -n is the reciprocal of an.

STUDY TIP

Mathematics often builds on concepts that have already been studied. Try to get in the regular habit of reviewing topics from earlier parts of the text.

Negative Integers as Exponents So far we have defined exponents that are whole numbers. For example,

50 = 1 and 23 = 2 # 2 # 2 = 8. What if an exponent is a negative integer? To answer this question, consider Table 5.4, which shows values for decreasing powers of 2. Note that each time the exponent on 2 decreases by 1, the resulting value is divided by 2.

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EXAMPLE 1 Evaluating negative exponents

Simplify each expression. (a) 2-3 (b) 7-1 (c) x -2 (d) (x + y)-8

Solution

(a) Because a�n = 1

an , 2�3 =

23 =

2 # 2 # 2 = 1

8 .

(b) 7�1 = 1

71 =

x 2

(d) (x + y)�8 = 1

(x + y)8

Now Try Exercises 7, 19, 25(b)

TECHNOLOGY NOTE

Negative Exponents Calculators can be used to evaluate negative exponents. The figure shows how a graphing cal- culator evaluates the expressions in parts (a) and (b) of Example 1.

2^(�3)�Frac 1/8

7^(�1)�Frac 1/7

CALCULATOR HELP To use the fraction feature (Frac), see Appendix A (pages AP-1 and AP-2).

The rules for exponents discussed in this chapter so far also apply to expressions hav- ing negative exponents. For example, we can apply the product rule, am # an = am + n, as follows.

T Add

2-3 # 22 = 2-3 + 2 = 2-1 = 1 2

We can check this result by evaluating the expression without using the product rule.

2-3 # 22 = 1 23

# 22 = 1 8

# 4 = 4 8 =

EXAMPLE 2 Using the product rule with negative exponents

Evaluate each expression. (a) 52 # 5-4 (b) 3-2 # 3-1

Solution T

Add

(a) 52 # 5�4 = 52 + (�4) = 5�2 = 1 52

= 1

(b) 3-2 # 3-1 = 3-2 + (-1) = 3-3 = 1 33

= 1

27 Now Try Exercise 9

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EXAMPLE 3 Using the rules of exponents

Simplify the expression. Write the answer using positive exponents. (a) x 2 # x -5 (b) ( y3)-4 (c) (rt)-5 (d) (ab)-3(a -2b)3

Solution (a) Using the product rule, am # an = am�n, gives

x 2 # x�5 = x 2 + (�5) = x�3 = 1 x 3

(b) Using the power rule, (am)n = amn, gives

( y3)�4 = y3(�4) = y�12 = 1

y12 .

(rt)�5 = r�5t�5 = 1

r 5 # 1

t 5 =

r 5t 5 .

This expression could also be simplified as follows.

(rt)-5 = 1

(rt)5 =

r 5t 5

(d) (ab)-3(a -2b)3 = a -3b -3a -6b 3

= a -3 + (-6)b -3 + 3

= a -9b 0

= 1

a9 # 1

= 1

Now Try Exercises 21, 27, 29(a)

The Quotient Rule Consider the division problem

32 =

3 # 3 # 3 # 3 3 # 3 =

3 3

# 3 3

# 3 # 3 = 1 # 1 # 32 = 32. Because there are two more 3s in the numerator than in the denominator, the result is

T Subtract

34 - 2 = 32.

That is, to divide exponential expressions having the same base, subtract the exponent of the denominator from the exponent of the numerator and keep the same base. This rule is called the quotient rule, which we express in symbols as follows.

THE QUOTIENT RULE

For any nonzero number a and integers m and n,

an = am - n.

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EXAMPLE 4 Using the quotient rule

Simplify each expression. Write the answer using positive exponents.

(a) 43

45 (b)

6a7

3a4 (c)

xy7

x 2y5

Solution

T Subtract

(a) 43

45 = 43 - 5 = 4-2 =

42 =

(b) 6a7

3a4 =

3 # a7

a4 = 2a7 - 4 = 2a3

x 2y5 =

x 1

x 2 # y7

y5 = x 1 - 2y7 - 5 = x -1y2 =

Now Try Exercises 13(b), 31(b), 33(a)

MAKING CONNECTIONS

The Quotient Rule and Simplifying Quotients

Some quotients can be simplified mentally. Because

x 5

x 3 =

x # x # x # x # x x # x # x ,

the quotient x 5

x 3 has five factors of x in the numerator and three factors of x in the denominator.

There are two more factors of x in the numerator than in the denominator, 5 - 3 = 2, so this expression simplifies to x 2. Similarly,

x 3

x 5 =

x # x # x x # x # x # x # x

has two more factors of x in the denominator than in the numerator. This quotient x 3

x 5 sim-

plifies to 1 x 2

. Use this technique to simplify the expressions

z4 ,

a8 , and

x 6y2

x 3y7 .

Other Rules for Exponents Other rules can be used to simplify expressions with negative exponents.

QUOTIENTS AND NEGATIVE EXPONENTS

The following three rules hold for any nonzero real numbers a and b and positive integers m and n.

1. 1

a -n = an 2.

a -n

b -m =

b m

an 3. aa

b b-n = ab

a bn

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We demonstrate the validity of these rules as follows.

1. 1

a -n =

= 1 # a n

1 = an

2. a -n

b -m =

b m

= 1

an # b m

1 =

b m

3. aa b b-n = a -n

b -n =

b n

= 1

an # b n

1 =

b n

an = ab

a bn

EXAMPLE 5 Working with quotients and negative exponents

Simplify each expression. Write the answer using positive exponents.

(a) 1

2-5 (b)

3-3

4-2 (c)

5x -4y2

10x 2y -4 (d) a 2

z2 b-4

Solution

(a) 1

2�5 = 25 = 2 # 2 # 2 # 2 # 2 = 32 (b) 3

�3

4�2 =

33 =

10x 2y�4 =

y2y4

2 x 2x 4 =

2 x 6 (d) a 2

z2 b�4 = az2

2 b4 = z8

24 =

Now Try Exercises 15(b), 17, 37, 47

The rules for natural number exponents that are summarized in Section 5.1 on page 303 also hold for integer exponents. Additional rules for integer exponents are summarized as follows.

RULES FOR INTEGER EXPONENTS

The following rules hold for nonzero real numbers a and b, and positive integers m and n.

Description Rule Example

Negative Exponents (1) a -n = 1

an 9-2 =

92 =

The Quotient Rule am

an = am - n

2-2 = 23 - (-2) = 25

Negative Exponents (2) 1

a -n = an

7-5 = 75

Negative Exponents (3) a -n

b -m =

b m

4-3

2-5 =

Negative Exponents (4) aa b b - n = ab

a bn a1

5 b - 2 = a5

1 b2 = 25

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Scientific Notation Powers of 10 are important because they are used in science to express numbers that are either very small or very large in absolute value. Table 5.5 lists some powers of 10. Note that if the power of 10 decreases by 1, the result decreases by a factor of 110, or equivalently, the decimal point is moved one place to the left. Table 5.6 shows the names of some important powers of 10.

READING CHECK

• What kinds of numbers are often expressed in scientific notation?

TABLE 5.5 Powers of 10

Power of 10 Value

103 1000

102 100

101 10

100 1

10-1 110 = 0.1

10-2 1100 = 0.01

10-3 11000 = 0.001

TABLE 5.6 Important Powers of 10

Power of 10 Name

1012 Trillion

109 Billion

106 Million

103 Thousand

10-1 Tenth

10-2 Hundredth

10-3 Thousandth

10-6 Millionth

Recall that numbers written in decimal notation are sometimes said to be in standard form. Decimal numbers that are either very large or very small in absolute value can be expressed in scientific notation.

TECHNOLOGY NOTE

Powers of 10 Calculators make use of sci- entific notation, as illustrated in the accompanying figure. The letter E denotes a power of 10. That is,

2.5E13 = 2.5 : 1013 and 5E�6 = 5 : 10�6.

25000000000000 2.5E13

.000005 5E�6

Note: The calculator has been set in scientific mode.

REAL-WORLD CONNECTION As mentioned in A Look Into Math for this section, the distance to the planet WASP-17b is about 5,880,000,000,000,000 miles. This distance can be written in scientific notation as 5.88 * 1015 because

5,880,000,000,000,000 = 5.88 * 1015. 8 15 decimal places

The 1015 indicates that the decimal point in 5.88 should be moved 15 places to the right. A typical virus is about 0.00000468 inch in diameter, which can be written in scientific

notation as 4.68 * 10-6 because

0.00000468 = 4.68 * 10-6. (1)1* 6 decimal places

The 10�6 indicates that the decimal point in 4.68 should be moved 6 places to the left. The following definition provides a more complete explanation of scientific notation.

SCIENTIFIC NOTATION

A real number a is in scientific notation when a is written in the form b * 10n, where 1 … 0b 0 6 10 and n is an integer.

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EXAMPLE 6 Converting scientific notation to standard form

Write each number in standard form. (a) 5.23 * 104 (b) 8.1 * 10-3 (c) 6 * 10-2

Solution (a) The positive exponent 4 indicates that the decimal point in 5.23 is to be moved 4 places

to the right.

5.23 * 104 = 5. 2 3 0 0. = 52,300 1 2 3 4

(b) The negative exponent - 3 indicates that the decimal point in 8.1 is to be moved 3 places to the left.

8.1 * 10�3 = 0. 0 0 8. 1 = 0.0081 1 2 3

Now Try Exercises 61, 63

The following steps can be used for writing a positive number a in scientific notation.

NOTE: The scientific notation for a negative number a is the opposite of the scientific nota- tion of 0a 0 . For example, 450 = 4.5 * 102 and - 450 = - 4.5 * 102.

WRITING A POSITIVE NUMBER IN SCIENTIFIC NOTATION

For a positive, rational number a expressed as a decimal, if 1 … a 6 10, then a = a * 100. Otherwise, use the following process to write a in scientific notation.

1. Move the decimal point in a until it becomes a number b such that 1 … b 6 10. 2. Let the positive integer n be the number of places the decimal point was moved. 3. Write a in scientific notation as follows. • If a Ú 10, then a = b * 10n. • If a 6 1, then a = b * 10-n.

EXAMPLE 7 Writing a number in scientific notation

Write each number in scientific notation. (a) 308,000,000 (U.S. population in 2010) (b) 0.001 (Approximate time in seconds for sound to travel one foot)

Solution (a) Move the assumed decimal point in 308,000,000 eight places to obtain 3.08.

3. 0 8 0 0 0 0 0 0.

1 2 3 4 5 6 7 8

Since 308,000,000 Ú 10, the scientific notation is 3.08 * 108.

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Numbers in scientific notation can be multiplied by applying properties of real numbers and properties of exponents.

(6 * 104) # (3 * 103) = (6 # 3) * (104 # 103) Properties of real numbers = 18 * 107 Product rule = 1.8 * 108 Scientific notation

Division can also be performed with scientific notation.

6 * 104

3 * 103 =

6 3

* 104

103 Property of fractions

= 2 * 101 Quotient rule

These results are supported in Figure 5.16, where the calculator is in scientific mode.

In the next example we show how to use scientific notation in an application.

CALCULATOR HELP To display numbers in scientific notation, see Appendix A (page AP-2).

Figure 5.16

(6�10^4)(3�10^3)

1.8E8 (6�10^4)/(3�10^3 )

2E1

EXAMPLE 8 Analyzing the cost of Internet advertising

In 2009, a total of $2.38 * 1010 was spent on Internet advertising in the United States. At that time the population of the United States was 3.05 * 108. Determine how much was spent per person on Internet advertising. (Source: New York Times.)

Solution To determine the amount spent per person, divide $2.38 * 1010 by 3.05 * 108.

2.38 * 1010

3.05 * 108 =

2.38

3.05 * 1010 - 8 � 0.78 * 102 = 78

In 2009, about $78 per person was spent on Internet advertising.

Now Try Exercise 97

CRITICAL THINKING

Estimate the number of seconds that you have been alive. Write your answer in scientific notation.

(b) Move the decimal point in 0.001 three places to obtain 1.

0. 0 0 1.

1 2 3

Since 0.001 6 1, the scientific notation is 1 * 10�3.

Now Try Exercises 75, 79

5.5 Putting It All Together

For the rules for integer exponents in this table, assume that a and b are nonzero real num- bers and that m and n are integers.

CONCEPT EXPLANATION EXAMPLES

Negative Integer Exponents a

-n = 1

an 2-4 =

24 =

16 , a -8 =

a8 , and

(xy)-2 = 1

(xy)2 =

x 2y2

continued on next page

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CONCEPT EXPLANATION EXAMPLES

Quotient Rule am

an = am - n 7

74 = 72 - 4 = 7-2 =

72 =

49 and

x 6

x 3 = x 6 - 3 = x 3

Quotients and Negative Integer Exponents

1. 1

a -n = an

2. a -n

b -m =

b m

3. aa b b-n = ab

a bn

1. 1

5-2 = 52 = 25

2. x -4

y -2 =

x 4

3. a2 3 b-3 = a3

2 b3 = 33

23 =

Scientific Notation Write a as b * 10n, where 1 … 0b 0 6 10 and n is an integer. 23,500 = 2.35 * 104,0.0056 = 5.6 * 10-3, and

1000 = 1 * 103

5.5 Exercises

CONCEPTS AND VOCABULARY

Exercises 1–5: Complete the given rule for integer expo- nents m and n, where a and b are nonzero real numbers.

1. a -n = 2. 1

a -n =

3. am

an = 4.

a -n

b -m =

5. aa b b-n =

6. To write a positive number a in scientific notation as b * 10n, the number b must satisfy _____.

NEGATIVE EXPONENTS

Exercises 7–18: Simplify the expression.

7. (a) 4-1 (b) a1 3 b-2

8. (a) 6-2 (b) 2.5-1

9. (a) 23 # 2-2 (b) 10-1 # 10-2

10. (a) 3-4 # 32 (b) 104 # 10-2 11. (a) 3-2 # 3-1 # 3-1 (b) (23)-1 12. (a) 2-3 # 25 # 2-4 (b) (3-2)-2

13. (a) (3243)-1 (b) 45

14. (a) (2-232)-2 (b) 55

15. (a) 19

17 (b)

4-3

16. (a) - 64

6 (b)

6-2

17. (a) 5-2

5-4 (b) a2

7 b-2

18. (a) 7-3

7-1 (b) a3

4 b-3

continued from previous page

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Exercises 19–50: Simplify the expression. Write the answer using positive exponents.

19. (a) x -1 (b) a -4

20. (a) y -2 (b) z-7

21. (a) x -2 # x -1 # x (b) a -5 # a -2 # a -1 22. (a) y -3 # y4 # y -5 (b) b 5 # b -3 # b -6 23. (a) x 2y -3x -5y6 (b) (xy)-3

24. (a) a -2b -6b 3a -1 (b) (ab)-1

25. (a) (2t)-4 (b) (x + 1)-7

26. (a) (8c)-2 (b) (a + b)-9

27. (a) (a -2)-4 (b) (rt 3)-2

28. (a) (4 x 3)-3 (b) (xy -3)-2

29. (a) (ab)2(a2)-3 (b) x 4

x 2

30. (a) (x 3)-2(xy)4y -5 (b) y9

31. (a) a10

a -3 (b)

2z4

32. (a) b 5

b -2 (b)

12 x 2

24 x 7

33. (a) - 4 xy5

6 x 3y2 (b)

x -4

x -1

34. (a) 12a6b 2

8ab 3 (b)

y -2

y -7

35. (a) 10b -4

5b -5 (b) aa

b b3

36. (a) 8a -2

2a -3 (b) a2 x

y b5

37. (a) 6 x 2y -4

18 x -5y4 (b)

16a -3b -5

4a -8b

38. (a) m2n4

3m-5n4 (b)

7x -3y -5

x -3y -2

39. (a) 1

y -5 (b)

2t -3

40. (a) 1

z-6 (b)

10b -5

41. (a) 3a4

(2a -2)3 (b)

(2b 5)-3

4b -6

42. (a) (2 x 4)-2

5x -2 (b)

2y5

(3y -4)-2

43. (a) 1

(xy)-2 (b)

(a2b)-3

44. (a) 1

(ab)-1 (b)

(rt 4)-2

45. (a) (3m4n)-2

(2mn-2)3 (b)

( - 4 x 4y)2

(xy -5)-3

46. (a) (x 4y2)2

( - 2 x 2y -2)3 (b)

(m2n-6)-2

(4m2n-4)-3

47. (a) aa b b-2 (b) a u

4v b-1

48. (a) a2 x y b-3 (b) a5u

3v b-2

49. (a) a 3a4b 2ab -2

b-2 (b) a 4m4n 5m-3n2

b2 50. (a) a 2 x 4y2

3x 3y -3 b3 (b) a a -5b 2

2ab -2 b-2

51. Thinking Generally For positive integers m and n

show that an

am =

am - n .

52. Thinking Generally For positive integers m and n

show that a -n

a -m = am - n.

SCIENTIFIC NOTATION

Exercises 53–58: (Refer to Table 5.6.) Write the value of the power of 10 in words.

53. 103 54. 106

55. 109 56. 10-1

57. 10-2 58. 10-6

Exercises 59–70: Write the expression in standard form.

59. 2 * 103 60. 5 * 102

61. 4.5 * 104 62. 7.1 * 106

63. 8 * 10-3 64. 9 * 10-1

65. 4.56 * 10-4 66. 9.4 * 10-2

67. 3.9 * 107 68. 5.27 * 106

69. - 5 * 105 70. - 9.5 * 103

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Exercises 71–82: Write the number in scientific notation.

71. 2000 72. 11,000

73. 567 74. 9300

75. 12,000,000 76. 600,000

77. 0.004 78. 0.0008

79. 0.000895 80. 0.0123

81. - 0.05 82. - 0.934

Exercises 83–90: Evaluate the expression. Write the answer in standard form.

83. (5 * 103)(3 * 102)

84. (2.1 * 102)(2 * 104)

85. ( - 3 * 10-3)(5 * 102)

86. (4 * 102)(1 * 103)(5 * 10-4)

87. 4 * 105

2 * 102 88.

9 * 102

3 * 106

89. 8 * 10-6

4 * 10-3 90.

6.3 * 102

2 * 10-3

APPLICATIONS

91. Light-year The distance that light travels in 1 year is called a light-year. Light travels at 1.86 * 105 miles per second, and there are about 3.15 * 107 seconds in 1 year. (a) Estimate the number of miles in 1 light-year. (b) Except for the sun, Alpha Centauri is the near-

est star, and its distance is 4.27 light-years from Earth. Estimate its distance in miles. Write your answer in scientific notation.

92. Milky Way It takes 2 * 108 years for the sun to make one orbit around the Milky Way galaxy. Write this number in standard form.

93. Speed of the Sun (Refer to the two previous exer- cises.) Assume that the sun’s orbit in the Milky Way galaxy is circular with a diameter of 105 light-years. Estimate how many miles the sun travels in 1 year.

94. Distance to the Moon The moon is about 240,000 miles from Earth.

(a) Write this number in scientific notation. (b) If a rocket traveled at 4 * 104 miles per hour,

how long would it take for it to reach the moon? 95. Online Exploration In 1997, the creators of the

Internet search engine BackRub renamed it Google. This new name is a play on the word googol, which is a very large number. Look up a googol and write it in scientific notation.

96. Online Exploration An astronomical unit (AU) is based on the distance from Earth to the sun. Look up the distance in kilometers from Earth to the sun.

(a) Write an astronomical unit in standard form to the nearest million kilometers.

(b) Convert your rounded answer from part (a) to scientific notation.

97. Gross Domestic Product The gross domestic prod- uct (GDP) is the total national output of goods and services valued at market prices within the United States. The GDP of the United States in 2005 was $12,460,000,000,000. (Source: Bureau of Economic Analysis.)

(a) Write this number in scientific notation. (b) In 2005, the U.S. population was 2.98 * 108. On

average, how many dollars of goods and services were produced by each individual?

98. Average Family Net Worth A family refers to a group of two or more people related by birth, mar- riage, or adoption who reside together. In 2000, the average family net worth was $280,000, and there were about 7.2 * 107 families. Calculate the total family net worth in the United States in 2000. (Source: U.S. Census Bureau.)

WRITING ABOUT MATHEMATICS

99. Explain what a negative exponent is and how it is different from a positive exponent. Give an example.

100. Explain why scientific notation is helpful for writ- ing some numbers.

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Working with Real DataGroup Activity

Water in a Lake East Battle Lake in Minnesota covers an area of about 1950 acres, or 8.5 * 107 square feet, and its average depth is about 3.2 * 101 feet. (a) Estimate the cubic feet of water in the lake. (Hint:

volume = area * average depth.) (b) One cubic foot of water equals about 7.5 gallons.

How many gallons of water are in this lake?

(c) The population of the United States is about 3.1 * 108, and the average American uses about 1.5 * 102 gallons of water per day. Could this lake supply the American population with water for 1 day?

Directions: Form a group of 2 to 4 people. Select a person to record the group’s responses for this activity. All members of the group should work cooperatively to answer the questions. If your instructor asks for your results, each member of the group should be prepared to respond.

5.6 Division of Polynomials Division by a Monomial ● Division by a Polynomial

A LOOK INTO MATH N The study of polynomials has occupied the minds of mathematicians for centuries. During the sixteenth century, Girolamo Cardano and other Italian mathematicians discovered how to solve higher degree polynomial equations. In this section we demonstrate how to divide polynomials. Division is often needed to factor polynomials and to solve polynomial equa- tions. (Source: H. Eves, An Introduction to the History of Mathematics.)

Division by a Monomial To add two fractions with like denominators, we use the property

d +

d =

a + b d

For example, 17 + 3 7 =

1 + 3 7 =

4 7.

To divide a polynomial by a monomial, we use the same property, only in reverse. That is,

a + b d

= a

d +

d .

Note that each term in the numerator is divided by the monomial in the denominator. The next example shows how to divide a polynomial by a monomial.

Girolamo Cardano (1501–1576)

EXAMPLE 1 Dividing a polynomial by a monomial

Divide.

(a) a5 + a3

a2 (b)

5x 4 + 10x 10x

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Solution

(a) a5 � a3

a2 =

a2 �

a2 = a5 - 2 + a3 - 2 = a3 + a

(b) 5x 4 � 10x

10x =

5x 4

10x �

10x

10x =

x 3

2 + 1

6y =

3y2

6y �

6y =

2 +

3 -

2 y

Now Try Exercises 17, 19, 21

MAKING CONNECTIONS

Division and Simplification

A common mistake made when dividing expressions is to “cancel” incorrectly. Note in Example 1(b) that

5x 4 + 10x 10x

3 5x 4 + 10x

10x .

The monomial must be divided into every term in the numerator.

When dividing two natural numbers, we can check our work by multiplying. For exam- ple, 105 = 2, and we can check this result by finding the product 5 # 2 = 10. Similarly, to check

a5 � a3

a2 = a3 � a

in Example 1(a) we can multiply a2 and a3 � a.

a2(a3 � a) = a2 # a3 + a2 # a Distributive property = a5 � a3 ✓ It checks.

EXAMPLE 2 Dividing and checking

Divide the expression 8 x 3 - 4 x 2 + 6 x

2 x 2 and then check the result.

Solution Be sure to divide 2 x 2 into every term in the numerator.

8 x 3 � 4 x 2 � 6 x

2 x 2 =

8 x 3

2 x 2 �

4 x 2

2 x 2 �

6 x

2 x 2 = 4 x - 2 +

3 x

Check:

2 x 2a4 x - 2 + 3 x b = 2 x 2 # 4 x - 2 x 2 # 2 + 2 x 2 # 3

= 8 x 3 - 4 x 2 + 6 x ✓

Now Try Exercise 23

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EXAMPLE 3 Finding the length of a rectangle

The rectangle in Figure 5.17 has an area A = x 2 + 2 x and width x. Write an expression for its length l in terms of x.

STUDY TIP

Do you have enough time to study your notes and complete your assign- ments? One way to man- age your time is to make a list of your time com- mitments and determine the amount of time that each activity requires. Remember to include time for eating, sleeping, and relaxing!

Figure 5.17

x A = x2 + 2x

Solution The area A of a rectangle equals length l times width w, or A = lw. Solving for l gives

l = A w

Thus to find the length of the given rectangle, divide the area by the width.

l = x 2 + 2 x

x =

x 2

x +

2 x x

= x + 2

The length of the rectangle is x + 2. The answer checks because x(x + 2) = x 2 + 2 x.

Now Try Exercise 49

To check this result, we find the product of the quotient and divisor and then add the remain- der. Because 67 # 4 + 3 = 271, the answer checks. The quotient and remainder can also be expressed as 6734. Division of polynomials is similar to long division of natural numbers.

Division by a Polynomial To understand division by a polynomial better, we first need to review some terminology related to long division of natural numbers. To compute 271 , 4, we complete long divi- sion as follows.

67 R 3

4�271 24

Quotient h vRemainder Divisorh v Dividend

EXAMPLE 4 Dividing polynomials

Divide 6 x 2 + 13x + 3

3x + 2 and check.

Solution Begin by dividing the first term of 3x + 2 into the first term of 6 x 2 + 13x + 3. That is, divide 3x into 6 x 2 to obtain 2 x. Then find the product of 2 x and 3x + 2, or 6 x 2 + 4 x, place it below 6 x 2 + 13x, and subtract. Bring down the 3.

2x 3x + 2�6 x 2 + 13x + 3

6 x 2 + 4 x 9x + 3

6 x 2

3x = 2 x

2 x(3x + 2) = 6 x 2 + 4 x Subtract: 13x - 4 x = 9x. Bring down the 3.

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346 CHAPTER 5 POLYNOMIALS AND EXPONENTS

In the next step, divide 3x into the first term of 9x + 3 to obtain 3. Then find the product of 3 and 3x + 2, or 9x + 6, place it below 9x + 3, and subtract.

2 x + 3 3x + 2�6 x 2 + 13x + 3

6 x 2 + 4 x 9x + 3 9x + 6

- 3

9x 3x

= 3

3(3x + 2) = 9x + 6 Subtract: 3 - 6 = - 3.

The quotient is 2 x + 3 with remainder - 3. This result can also be written as

2 x + 3 + - 3

3x + 2 , Quotient +

Remainder

Divisor

in the same manner that 67 R 3 was written as 6734. Check polynomial division by adding the remainder to the product of the divisor and

the quotient. That is,

(Divisor)(Quotient) + Remainder = Dividend.

For this example, the equation becomes

(3x � 2)(2 x � 3) + (�3) = 3x # 2 x + 3x # 3 + 2 # 2 x + 2 # 3 � 3 = 6 x 2 + 9x + 4 x + 6 - 3 = 6 x 2 � 13x � 3. ✓ It checks.

Now Try Exercise 27

READING CHECK

• How can you check a polynomial division problem?

EXAMPLE 5 Dividing polynomials having a missing term

Simplify (3x 3 + 2 x - 4) , (x - 2).

Solution Because the dividend does not have an x 2-term, insert 0x 2 as a “place holder.” Then begin by dividing x into 3x 3 to obtain 3x 2.

3x 2

x - 2�3x 3 + 0 x 2 + 2 x - 4 3x 3 - 6 x 2

6 x 2 + 2 x

3x 3

x = 3x 2

3x 2(x - 2) = 3x 3 - 6 x 2

Subtract 0x 2 - ( - 6 x 2) = 6 x 2. Bring down 2x.

In the next step, divide x into 6 x 2.

3x 2 + 6 x x - 2�3x 3 + 0 x 2 + 2 x - 4

3x 3 - 6 x 2

6 x 2 + 2 x 6 x 2 - 12 x

14 x - 4

6 x 2

x = 6 x

6 x(x - 2) = 6 x 2 - 12 x Subtract: 2 x - ( - 12 x) = 14 x. Bring down - 4.

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Now divide x into 14 x.

3x 2 + 6 x + 14 x - 2�3x 3 + 0 x 2 + 2 x - 4

3x 3 - 6 x 2

6 x 2 + 2 x 6 x 2 - 12 x

14 x - 4 14 x - 28

The quotient is 3x 2 + 6 x + 14 with remainder 24. This result can also be written as

3x 2 + 6 x + 14 + 24

x - 2 .

Now Try Exercise 37

EXAMPLE 6 Dividing when the divisor is not linear

Divide x 3 - 3x 2 + 3x + 2 by x 2 + 1.

Solution Begin by writing x 2 + 1 as x 2 + 0x + 1.

x - 3 x 2 + 0x + 1�x 3 - 3x 2 + 3x + 2

x 3 + 0x 2 + x - 3x 2 + 2 x + 2 - 3x 2 + 0x - 3

2 x + 5

The quotient is x - 3 with remainder 2 x + 5. This result can also be written as

x - 3 + 2 x + 5 x 2 + 1

Now Try Exercise 41

5.6 Putting It All Together

Division by a Monomial Use the property

a + b d

= a

d +

d .

Be sure to divide the denominator into every term in the numerator.

2 x 3 + 4 x 2 x 2

= 2 x 3

2 x 2 +

4 x

2 x 2 = x +

2 x

and

a2 - 2a 4a

= a2

4a -

4a =

4 -

CONCEPT EXPLANATION EXAMPLES

continued on next page

14 x x

= 14

14(x - 2) = 14 x - 28 Subtract: - 4 - ( - 28) = 24.

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CONCEPT EXPLANATION EXAMPLES

Division by a Polynomial

Is done similarly to the way long divi- sion of natural numbers is performed

If either the divisor or the dividend is missing a term, be sure to insert as a “place holder” the missing term with coefficient 0.

Divide x 2 + 3x + 3 by x + 1.

x + 2 x + 1�x 2 + 3x + 3

x 2 + x 2 x + 3 2 x + 2

The quotient is x + 2 with remainder 1, which can be expressed as

x + 2 + 1

x + 1 .

Checking a Result Dividend = (Divisor)(Quotient) + Remainder

When x 2 + 3x + 3 is divided by x + 1, the quotient is x + 2 with remainder 1. Thus

(x + 1)(x + 2) + 1 = x 2 + 3x + 3,

and the answer checks.

continued from previous page

CHAPTER 5 POLYNOMIALS AND EXPONENTS348

CONCEPTS AND VOCABULARY

1. a + bd = 2. a + b - c

d =

3. When dividing a polynomial by a monomial, the mono- mial must be divided into every of the polynomial.

4. (True or False?) The expressions 5x 2 + 2 x

2 x and 5x 2 + 1

are equal.

5. (True or False?) The expressions 5x 2 + 2 x

2 x and 5x 2 2 x are

equal.

6. Because 379 = 4 with remainder 1, it follows that 37 = # + .

7. Because 2 x 3 - x + 5 divided by x + 1 equals 2 x 2 - 2 x + 1 with remainder 4, it follows that 2 x 3 - x + 5 = _____ # _____ + _____.

8. When dividing 2 x 3 + 3x - 1 by x - 1, insert

into the dividend as a “place holder” for the missing x 2-term.

DIVISION BY A MONOMIAL

Exercises 9–16: Divide and check.

9. 6 x 2

3x 10.

- 5x 2

10x 4

11. z4 + z3

z 12.

t 3 - t t

13. a5 - 6a3

2a3 14.

b 4 - 4b 4b 2

15. y + 6y2

3y3 16.

8z2 - z 4z2

Exercises 17–26: Divide.

17. 4 x - 7x 4

x 2 18.

1 + 6 x 4

3x 3

19. 6y2 + 3y

3y3 20.

5z2 - 10z3

5z4

5.6 Exercises

21. 9x 4 - 3x + 6

3x 22.

y3 - 4y + 6 y

23. 12y4 - 3y2 + 6y

3y2 24.

2 x 2 - 6 x + 9 12 x

25. 15m4 - 10m3 + 20m2

5m2 26.

n8 - 8n6 + 4n4

2n5

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47. Thinking Generally If the quotient in a polynomial division problem is an integer, what must be true about the degrees of the dividend and divisor?

48. Thinking Generally If the quotient in a polynomial division problem is a polynomial of degree 1, what must be true about the degrees of the dividend and divisor?

APPLICATIONS

Exercises 49 and 50: Area of a Rectangle The area of a rectangle and its width are given. Find an expression for the length l.

49. 50.

51. Volume of a Box The volume V of a box is 2 x 3 + 4 x 2, and the area of its bottom is 2 x 2. Find the height of the box in terms of x. Make a possible sketch of the box, and label the length of each side.

52. Area of a Triangle A triangle has height h and area A = 2h 2 - 4h. Find its base b in terms of h. Make a possible sketch of the triangle, and label the height and base. (Hint: A = 12 bh.)

WRITING ABOUT MATHEMATICS

53. Suppose that one polynomial is divided into another polynomial and the remainder is 0. What does the product of the divisor and quotient equal? Explain.

54. A student simplifies the expression 4 x 3 - 1 4 x 2

to x - 1. Explain the student’s error.

2x A = 8x2

l l

x – 1 A = x2 – 1

27. 2 x 2 - 3x + 1

x - 2 28.

4 x 2 - x + 3 x + 2

Exercises 27–34: Divide and check.

29. x 2 + 2 x + 1

x + 1 30.

4 x 2 - 4 x + 1 2 x - 1

31. x 3 - x 2 + x - 2

x - 1 32.

2 x 3 + 3x 2 + 3x - 1 2 x + 1

33. x 3 + x 2 - 7x + 2

x - 2

34. x 3 + x 2 - 2 x + 12

x + 3

Exercises 35– 46: Divide.

35. 4 x 3 - 3x 2 + 7x + 3

4 x + 1

36. 10x 3 - x 2 - 17x - 7

5x + 2

37. x 3 - x + 2

x - 2

38. 6 x 3 + 8 x 2 + 4

3x + 4

39. (3x 3 + 2) , (x - 1)

40. ( - 3x 3 + 8 x 2 + x) , (3x + 4)

41. (x 3 + 3x 2 + 1) , (x 2 + 1)

42. (x 4 - x 3 + x 2 - x + 1) , (x 2 - 1)

43. x 3 + 1

x 2 - x + 1 44.

4 x 3 + 3x + 2 2 x 2 - x + 1

45. x 3 + 8 x + 2

46. x 4 - 16 x - 2

Checking Basic ConceptsSECTIONS5.5 and 5.6

1. Simplify each expression. Write the result with positive exponents.

(a) 9-2 (b) 3x -3

6 x 4 (c) (4ab -4)-2

2. Simplify each expression. Write the result with positive exponents.

(a) 1

z-5 (b)

x -3

y -6 (c) a 3

x 2 b-3

3. Write each number in scientific notation. (a) 45,000 (b) 0.000234 (c) 0.01

4. Write each expression in standard form. (a) 4.71 * 104 (b) 6 * 10-3

5. Simplify 25a 4 - 15a 3 5a 3

6. Divide 3x 2 - x - 4 by x - 1. State the quotient and remainder.

7. Divide x 4 + 2 x 3 - 2 x 2 - 5x - 2 by x 2 - 3. State the quotient and remainder.

8. Distance to the Sun The distance to the sun is

approximately 93 million miles. (a) Write this distance in scientific notation. (b) Light travels at 1.86 * 105 miles per second.

How long does it take for the sun’s light to reach Earth?

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CHAPTER 5 Summary SECTION 5.1 . RULES FOR EXPONENTS

Bases and Exponents The expression b n has base b and exponent n and equals the expression b # b # b # g # b, when n is a natural number.u

n times

Example: 23 has base 2 and exponent 3 and equals 2 # 2 # 2 = 8. Evaluating Expressions When evaluating expressions, evaluate exponents before performing addition, subtraction, multiplication, division, or negation. In general, operations within paren- theses should be evaluated before using the order of operations.

1. Evaluate exponents. 2. Perform negation. 3. Do multiplication and division from left to right. 4. Do addition and subtraction from left to right.

Example: - 32 + 3 # 4 = - 9 + 3 # 4 = - 9 + 12 = 3 Zero Exponents For any nonzero number b, b 0 = 1. Note that 00 is undefined.

Examples: 50 = 1 and ax y b0 = 1, where x and y are nonzero.

Product Rule For any real number a and natural numbers m and n,

am # an = am + n. Examples: 34 # 32 = 36 and x 3x 2x 4 = x 9 Power Rules For any real numbers a and b and natural numbers m and n,

(am)n = amn, (ab)n = anb n, and aa b bn = an

b n , b � 0.

Examples: (x 2)3 = x 6, (3x)4 = 34x 4 = 81x 4, and a2 y b3 = 23

y3 =

SECTION 5.2 . ADDITION AND SUBTRACTION OF POLYNOMIALS

Terms Related to Polynomials Monomial A number, variable, or product of numbers and variables

raised to natural number powers

Degree of a Monomial Sum of the exponents of the variables

Coefficient of a Monomial The number in a monomial

Example: The monomial - 3x 2y3 has degree 5 and coefficient - 3.

Polynomial A monomial or the sum of two or more monomials

Term of a Polynomial Each monomial is a term of the polynomial.

Binomial A polynomial with two terms

Trinomial A polynomial with three terms

Degree of a Polynomial The degree of the term with highest degree

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Opposite of a Polynomial The opposite is found by negating each term.

Example: 2 x 3 - 4 x + 5 is a trinomial with degree 3. Its opposite is - 2 x 3 + 4 x - 5.

Like Terms Two monomials with the same variables raised to the same powers

Examples: 3xy2 and - xy2 are like terms.

5x 3 and 3x 3 are like terms.

5x 2 and 5x are unlike terms.

Addition of Polynomials Combine like terms, using the distributive property.

Example: (2 x 2 - 4 x) + ( - x 2 - x) = (2 - 1)x 2 + ( - 4 - 1)x = x 2 - 5x

Subtraction of Polynomials Add the first polynomial to the opposite of the second polynomial.

Example: (4 x 4 - 5x) - (7x 4 + 6 x) = (4 x 4 - 5x) + ( - 7x 4 - 6 x) = (4 - 7)x 4 + ( - 5 - 6)x = - 3x 4 - 11x

SECTION 5.3 . MULTIPLICATION OF POLYNOMIALS

Multiplication of Monomials Use the commutative property and the product rule.

Examples: - 2 x 3 # 3x 2 = - 2 # 3 # x 3 # x 2 = - 6 x 5 (2 xy2)(3x 2y3) = 2 # 3 # x # x 2 # y2 # y3 = 6 x 3y5 c Assumed exponent of 1

Distributive Properties

a(b + c) = ab + ac and a(b - c) = ab - ac

Examples: 4 x(3x + 6) = 4 x # 3x + 4 x # 6 = 12 x 2 + 24 x ab(a2 - b 2) = ab # a2 - ab # b 2 = a3b - ab 3 Multiplication of Monomials and Polynomials Apply the distributive properties. Be sure to multiply every term in the polynomial by the monomial.

Example: - 2 x 2(4 x 2 - 5x - 3) = - 8 x 4 + 10x 3 + 6 x 2

Multiplication of Polynomials The product of two polynomials may be found by multiplying every term in the first polynomial by every term in the second polynomial. Be sure to combine like terms.

Examples: (x + 3)(2 x - 5) = 2 x 2 - 5x + 6 x - 15

= 2 x 2 + x - 15

(2 x + 1)(x 2 - 5x + 2) = 2 x 3 - 10x 2 + 4 x + x 2 - 5x + 2 = 2 x 3 - 9x 2 - x + 2

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SECTION 5.4 . SPECIAL PRODUCTS

Product of a Sum and Difference

(a + b)(a - b) = a2 - b 2

Examples: (x + 4)(x - 4) = x 2 - 16 (2r - 3t)(2r + 3t) = (2r)2 - (3t)2 = 4r 2 - 9t 2

Squaring Binomials

(a + b)2 = a2 + 2ab + b 2 and (a - b)2 = a2 - 2ab + b 2

Examples: (2 x + 1)2 = (2 x)2 + 2(2 x)1 + 12 = 4 x 2 + 4 x + 1 (z2 - 2)2 = (z2)2 - 2z2(2) + 22 = z4 - 4z2 + 4

Cubing Binomials To multiply (a + b)3, write it as (a + b)(a + b)2.

Example: (x + 4)3 = (x + 4)(x + 4)2

= (x + 4)(x 2 + 8 x + 16) Square the binomial. = x 3 + 8 x 2 + 16 x + 4 x 2 + 32 x + 64 Distributive property = x 3 + 12 x 2 + 48 x + 64 Combine like terms.

SECTION 5.5 . INTEGER EXPONENTS AND THE QUOTIENT RULE

Negative Integers as Exponents For any nonzero real number a and positive integer n,

a -n = 1

an .

Examples: 5-2 = 1

52 and x -4 =

x 4

The Quotient Rule For any nonzero real number a and integers m and n,

an = am - n.

Examples: 64

62 = 64 - 2 = 62 = 36 and

xy3

x 4y2 = x 1 - 4y3 - 2 = x -3y1 =

x 3

Other Rules For any nonzero real numbers a and b and positive integers m and n,

a -n = an,

a -n

b -m =

b m

an , and aa

b b-n = ab

a bn.

Examples: 1

4-3 = 43,

x -3

y -2 =

x 3 , and a4

5 b-2 = a5

4 b2

Scientific Notation A real number a written as b * 10n, where 1 … 0b 0 6 10 and n is an integer Examples: 2.34 * 103 = 2340 Move the decimal point 3 places to the right. 2.34 * 10-3 = 0.00234 Move the decimal point 3 places to the left.

SECTION 5.6 . DIVISION OF POLYNOMIALS

Division of a Polynomial by a Monomial Divide the monomial into every term of the polynomial.

Example: 5x 3 - 10x 2 + 15x

5x =

5x 3

5x -

10x 2

5x +

15x

5x = x 2 - 2 x + 3

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Division of a Polynomial by a Polynomial Division of polynomials is performed similarly to long division of natural numbers.

Example: Divide 2 x 3 + 4 x 2 - 3x + 1 by x + 1.

2 x 2 + 2 x - 5 x + 1�2 x 3 + 4 x 2 - 3x + 1

2 x 3 + 2 x 2

2 x 2 - 3x 2 x 2 + 2 x

- 5x + 1 - 5x - 5

The quotient is 2 x 2 + 2 x - 5 with remainder 6, which can be written as

2 x 2 + 2 x - 5 + 6

x + 1 .

CHAPTER 5 Review Exercises SECTION 5.1

Exercises 1– 6: Evaluate the expression.

1. 53 2. - 34

3. 4( - 2)0 4. 3 + 32 - 30

5. - 52

5 6. a - 5

5 b2

Exercises 7–24: Simplify the expression.

7. 62 # 63 8. 105 # 107 9. z4 # z5 10. y2 # y # y3 11. 5x 2 # 6 x 7 12. (ab 3)(a3b) 13. (25)2 14. (m4)5

15. (ab)3 16. (x 2y3)4

17. (xy)3(x 2y4)2 18. (a2b 9)0

19. (r - t)4(r - t)5 20. (a + b)2(a + b)4

21. a 3 x - yb2 22. ax + y2 b3

23. 2 x 2(3x - 5) 24. 3x(4 x + x 3)

SECTION 5.2

Exercises 25 and 26: Identify the degree and coefficient of the monomial.

25. 6 x 7 26. - x 2y3

Exercises 27–30: Determine whether the expression is a polynomial. If it is, state how many terms and variables the polynomial contains. Then state its degree.

27. 8y 28. 8 x 3 - 3x 2 + x - 5

29. a2 + 2ab + b 2 30. 1 xy

31. Add the polynomials vertically.

3x 2 + 4 x + 8 2 x 2 - 5x - 5

32. Write the opposite of 6 x 2 - 3x - 7.

Exercises 33– 40: Simplify.

33. (4 x - 3) + ( - x + 7)

34. (3x 2 - 1) - (5x 2 + 12)

35. (x 2 + 5x + 6) - (3x 2 - 4 x + 1)

36. (x 2 + 3x - 5) + (2 x 2 - 5x - 1)

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37. (a3 + 4a2) + (a3 - 5a2 + 7a)

38. (4 x 3 - 2 x + 6) - (4x 3 - 6)

39. (xy + y2) + (4y2 - 4 xy)

40. (7x 2 + 2 xy + y2) - (7x 2 - 2 xy + y2)

SECTION 5.3

Exercises 41–54: Multiply and simplify.

41. - x 2 # x 3 42. - (r 2t 3)(rt) 43. - 3(2t - 5) 44. 2y(1 - 6y)

45. 6 x 3(3x 2 + 5x)

46. - x(x 2 - 2 x + 9)

47. - ab(a2 - 2ab + b 2)

48. (a - 2)(a + 5)

49. (8 x - 3)(x + 2) 50. (2 x - 1)(1 - x)

51. ( y2 + 1)(2y + 1) 52. ( y2 - 1)(2y2 + 1)

53. (z + 1)(z2 - z + 1)

54. (4z - 3)(z2 - 3z + 1)

Exercises 55 and 56: Multiply the expression (a) geometrically and (b) symbolically.

55. z(z + 1) 56. 2 x(x + 2)

SECTION 5.4

Exercises 57–72: Multiply.

57. (z + 2)(z - 2) 58. (5z - 9)(5z + 9) 59. (1 - 3y)(1 + 3y) 60. (5x + 4y)(5x - 4y)

61. (rt + 1)(rt - 1)

62. (2m2 - n2)(2m2 + n2)

63. (x + 1)2 64. (4 x + 3)2

65. ( y - 3)2 66. (2y - 5)2

67. (4 + a)2 68. (4 - a)2

69. (x 2 + y2)2

70. (xy - 2)2

71. (z + 5)3

72. (2z - 1)3

Exercises 73 and 74: Use the product of a sum and a dif- ference to evaluate the expression.

73. 59 # 61 74. 22 # 18 SECTION 5.5

Exercises 75–82: Simplify the expression.

75. 9-1 76. 3-2

77. 43 # 4-2 78. 10-6 # 103

79. 1

6-2 80.

81. (3-1 22)-2 82. (2-4 53)0 1

Exercises 83–98: Simplify the expression. Write the answer using positive exponents.

83. z-2 84. y -4

85. a -4 # a2 86. x 2 # x -5 # x 87. (2t)-2 88. (ab 2)-3

89. (xy)-2(x -2y)-1 90. x 6

x 2

91. 4 x

2 x 4 92.

20x 5y3

30xy6

93. aa b b5 94. 4

t -4

95. (3m3n)-2

(2m2n-3)3 96. a x -4y2

3xy -3 b-2

97. ax y b-2 98. a3u

2v b-1

Exercises 99–102: Write the expression in standard form.

99. 6 * 102 100. 5.24 * 104

101. 3.7 * 10-3 102. 6.234 * 10-2

Exercises 103–106: Write the number in scientific notation.

103. 10,000 104. 56,100,000

105. 0.000054 106. 0.001

Exercises 107 and 108: Evaluate the expression. Write the result in standard form.

107. (4 * 102)(6 * 104) 108. 8 * 103

4 * 104

SECTION 5.6

Exercises 109–116: Divide and check.

109. 5x 2 + 3x

3x 110.

6b 4 - 4b 2 + 2 2b 2

111. 3x 2 - x + 2

x - 1 112.

9x 2 - 6 x - 2 3x + 2

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114. 2 x 3 - x 2 - 1

2 x - 1 115.

x 3 - x 2 - x + 1 x 2 + 1

116. x 4 + 3x 3 + 8 x 2 + 7x + 5

x 2 + x + 1

113. 4 x 3 - 11x 2 - 7x - 1

4 x + 1

APPLICATIONS

117. Heart Rate An athlete starts running and continues for 10 seconds. The polynomial 12 t

2 + 60 calculates the heart rate of the athlete in beats per minute t sec- onds after beginning the run, where t … 10. (a) What is the athlete’s heart rate when the athlete

first starts to run? (b) What is the athlete’s heart rate after 10 seconds? (c) What happens to the athlete’s heart rate while

the athlete is running?

118. Areas of Rectangles Find a monomial equal to the sum of the areas of the rectangles. Calculate this sum for x = 3 feet and y = 4 feet.

119. Area of a Rectangle Write a polynomial that gives the area of the rectangle. Calculate its area for z = 6 inches.

120. Area of a Square Find the area of the square whose sides have length x 2y.

x2y

121. Compound Interest If P dollars are deposited in an account that pays 6% annual interest, then the amount of money after 3 years is given by P(1 + 0.06)3. Find this amount when P = $700.

122. Volume of a Sphere The expression for the volume of a sphere with radius r is 43 pr

3. Find a polynomial that gives the volume of a sphere with radius x + 2. Leave your answer in terms of p.

x + 2

123. Height Reached by a Baseball A baseball is hit straight up. Its height h in feet above the ground after t seconds is given by t(96 - 16t). (a) Multiply this expression. (b) Evaluate both the expression in part (a) and the

given expression for t = 2. Interpret the result.

124. Rectangular Building A rectangular building has a perimeter of 1200 feet. (a) If one side of the building has length L, write

a polynomial expression that gives its area. (Be sure to multiply your expression.)

(b) Evaluate the expression in part (a) for L = 50 and interpret the answer.

125. Geometry Complete each part and verify that your answers are equal. (a) Find the area of the large square by multiplying

its length and width. (b) Find the sum of the areas of the smaller rect-

angles inside the large square.

x 5

126. Digital Picture A digital picture, including its border, is x + 4 pixels by x + 4 pixels, and the actual picture inside the border is x - 4 pixels by x - 4 pixels.

x + 4 x – 4

x + 4

x – 4

(a) Find a polynomial that gives the number of pix- els in the border.

(b) Let x = 100 and evaluate the polynomial.

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127. Federal Debt In 1990, the federal debt held by the public was $2.19 trillion, and the population of the United States was 249 million. Use scientific nota- tion to approximate the national debt per person. (Source: U.S. Department of the Treasury.)

128. Alcohol Consumption In 2007, about 239 million people in the United States were age 14 or older. They consumed, on average, 2.31 gallons of alcohol per person. Use scientific notation to estimate the total number of gallons of alcohol consumed by this age group. (Source: Department of Health and Human Services.)

1. Simplify each expression. (a) - 50 (b) - 92

2. Evaluate each expression by hand. (a) - 42 + 10 (b) 8-2

2-3 (d) - 3x 0

3. State how many terms and variables the polynomial 5x 2 - 3xy - 7y3 contains. Then state its degree.

4. Write the opposite of - x 3 + 4 x - 8.

Exercises 5–8: Simplify.

5. ( - 3x + 4) + (7x + 2)

6. ( y3 - 2y + 6) - (4y3 + 5)

7. (5x 2 - x + 3) - (4 x 2 - 2 x + 10)

8. (a3 + 5ab) + (3a3 - 3ab)

Exercises 9–16: Write the given expression with positive exponents.

9. 6y4 # 4y7 10. (a2b 3)2(ab 2) 11. x 7 # x -3 12. (a -1b 2)-3

13. ab(a2 - b 2) 14. a 3a2 2b -3

b-2 15.

12 xy4

6 x 2y 16. a 2

a + b b4

Exercises 17–22: Multiply and simplify.

24. Write 6.1 * 10-3 in standard form.

25. Write 5410 in scientific notation.

Exercises 26 and 27: Divide.

CHAPTER 5 Test Step-by-step test solutions are found on the Chapter Test Prep Videos available via the Video Resources on DVD, in , and on (search “RockswoldBeginAlg” and click on “Channels”).

17. 3x 2(4 x 3 - 6 x + 1)

19. (7y2 - 3)(7y2 + 3) 20. (3x - 2)2

18. (z - 3)(2z + 4)

21. (m + 3)3

22. ( y + 2)( y2 - 2y + 3)

23. Evaluate 78 # 82 using the product of a sum and a difference.

26. 9x 3 - 6 x 2 + 3x

3x 2 27.

x 3 + x 2 - x + 1 x + 2

28. Concert Tickets Tickets for a concert are sold for

$20 each. (a) Write a polynomial that gives the revenue from

selling t tickets. (b) Putting on the concert costs management $2000

to hire the band plus $2 for each ticket sold. What is the total cost of the concert if t tickets are sold?

(c) Subtract the polynomial that you found in part (b) from the polynomial that you found in part (a). What does this polynomial represent?

29. Areas of Rectangles Find a polynomial representing

the sum of the areas of two identical rectangles that have width 2 x and length 3x. Calculate this sum for x = 10 feet.

30. Volume of a Box Write a polynomial that repre- sents the volume of the box. Be sure to multiply your answer completely.

x + 6

x + 3 3x

31. Height Reached by a Golf Ball When a golf ball is hit into the air, its height in feet above the ground after t seconds is given by t(88 - 16t). (a) Multiply this expression. (b) Evaluate the expression in part (a) for t = 3.

Interpret the result.

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357CHAPTER 5 EXTENDED AND DISCOVERY EXERCISES

Exercises 1– 6: Arithmetic and Scientific Notation The product (4 * 103) * (2 * 102) can be evaluated as

(4 * 2) * (103 * 102) = 8 * 105,

and the quotient (4 * 103) , (2 * 102) can be evalu- ated as

4 * 103

2 * 102 =

2 *

103

102 = 2 * 101.

How would you evaluate (4 * 103) + (2 * 102)? How would you evaluate (4 * 103) - (2 * 102)? Make a con- jecture as to how numbers in scientific notation should be added and subtracted. Try your method on these prob- lems and then check your answers with a calculator set in scientific mode. Does your method work?

1. (4 * 103) + (3 * 103)

2. (5 * 10-2) - (2 * 10-2)

3. (1.2 * 104) - (3 * 103)

4. (2 * 102) + (6 * 101)

5. (2 * 10-1) + (4 * 10-2)

6. (2 * 10-3) - (5 * 10-2)

Exercises 7 and 8: Constructing a Box A box is con- structed from a rectangular piece of metal by cutting squares from the corners and folding up the sides. The square, cutout corners are x inches by x inches.

x x x

7. Suppose that the dimensions of the metal piece are 20 inches by 30 inches. (a) Write a polynomial that gives the volume of the

box. (b) Find the volume of the box for x = 4 inches.

8. Suppose that the metal piece is square with sides of length 25 inches. (a) Write a polynomial expression that gives the out-

side surface area of the box. (Assume that the box does not have a top.)

(b) Find this area for x = 3 inches.

Exercises 9–12: Calculators and Polynomials A graph- ing calculator can be used to help determine whether two polynomial expressions in one variable are equal. For example, suppose that a student believes that (x + 2)2 and x 2 + 4 are equal. Then the first two calculator tables shown demonstrate that the two expressions are not equal except for x = 0.

Y1�(X�2)2

X Y1 �3 1 �2 0 �1 1 0 4 1 9 2 16 3 25

Y1�X2�4

X Y1 �3 13 �2 8 �1 5 0 4 1 5 2 8 3 13

The next two calculator tables support the fact that (x + 1)2 and x 2 + 2 x + 1 are equal for all x.

Y1�(X�1)2

X Y1 �3 4 �2 1 �1 0 0 1 1 4 2 9 3 16

Y1�X2�2X�1

X Y1 �3 4 �2 1 �1 0 0 1 1 4 2 9 3 16

Use a graphing calculator to determine whether the first expression is equal to the second expression. If the expressions are not equal, multiply the first expression and simplify it.

9. 3x(4 - 5x), 12 x - 5x

10. (x - 1)2, x 2 - 1

11. (x - 1)(x 2 + x + 1), x 3 - 1

12. (x - 2)3, x 3 - 8

CHAPTER 5 Extended and Discovery Exercises

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358 CHAPTER 5 POLYNOMIALS AND EXPONENTS

CHAPTERS 1–5 Cumulative Review Exercises

Exercises 1 and 2: Evaluate each expression by hand.

1. (a) 18 - 2 # 5 (b) 42 , 7 + 2 2. (a) 21 - ( - 8) (b) - 73 , 1 - 149 2 Exercises 3 and 4: Solve the equation. Note that these equations may have no solutions, one solution, or infi- nitely many solutions.

3. (a) (x - 3) + x = 4 + x (b) 2(5x - 4) = 1 + 10x

4. (a) 2 + 6 x = 2(3x + 1) (b) 11x - 9 = - 31

5. Find the average speed of a car that travels 306 miles in 4 hours 30 minutes.

6. Write each value as a fraction in lowest terms. (a) 42% (b) 0.076

7. Graph the equation 4 x - 5y = 20.

8. Sketch a line with slope - 23 that passes through the point (1, 1).

9. Write the slope–intercept form for the line shown.

–2–3 –1 2 3

–2

–3

–1

10. Find the x-and y-intercepts for the graph of the equa- tion 2y = 3x - 6.

Exercises 11 and 12: Write the slope–intercept form of a line that satisfies the given information.

11. Parallel to 3x - 6y = 7, passing through (2, - 3) 12. Passing through ( - 2, - 5) and (1, 4)

Exercises 13–16: Solve the system of equations. Note that these systems may have no solutions, one solution, or infinitely many solutions.

Exercises 17 and 18: Shade the solution set for the system of inequalities.

13. 4 x + 3y = - 6 8 x + 6y = 12

14. x - 3y = 5 3x + y = 5

15. x + 4y = - 8 - 3x - 12y = 24

16. x - 5y = 30 2 x + y = - 6

17. x + y 6 3 y Ú x + 2

18. x - 2y 7 4 3x + y 6 6

19. Simplify the expression. (a) 3x 2 # 5x 3 (b) (x 3y)2(x 4y5)

20. Simplify. (a) (5x 2 - 3x + 4) - (3x 2 - 2 x + 1) (b) (7a3 - 4a2 - 5) + (5a3 + 4a2 + a)

21. Multiply and simplify. (a) (2 x + 3)(x - 7) (b) (y + 3)(y2 - 3y - 1)

22. Simplify the expression. Write the answer using posi- tive exponents. (a) x -5 # x 3 # x (b) a 2

x 3 b-3

6 x -2y (d) (xy -2)3(x -2y)-2

23. Write 24,000,000,000 in scientific notation.

24. Write 4.71 * 10-7 in standard form.

25. Divide.

(a) 8 x 3 - 2 x

2 x (b)

2 x 2 + x - 14 x + 3

26. Price Decrease If the price of a computer is reduced from $1200 to $900, find the percent change.

27. Mixing an Acid Solution How many milliliters of a 3% acid solution should be added to 400 milliliters of a 6% acid solution to dilute it to a 5% acid solution?

28. Surface Area of a Box Use the drawing of the box to write a polynomial that represents the area of each of the following.

x + 5

x + 2 2x

(a) The bottom (b) The front (c) The right side (d) All six sides

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