Section 5.1 Practice Exercises

Vocabulary and Key Concepts

1.

1. A(n) is used to show repeated multiplication of the base.

2. For b ≠ 0, the expression b0 is defined to be .

3. For b ≠ 0, the expression b−n

is defined as .

4. A number expressed in the form a × 10n, where 1 ≤ |a| < 10 and n is an integer is

said to be written in .

Concept 1: Simplifying Expressions with Exponents

2. Write the expressions in expanded form and simplify.

3. Write the expressions in expanded form and simplify.

For Exercises 4–9, write an example of each property. (Answers may vary.)

4. bn · b

m = b

n + m

5. (ab)n = a

nb

n

6. (bn)m = b

nm

7.

8.

9. b0 = 1 (b ≠ 0)

For Exercises 10–28, simplify. (See Example 1.)

10.

11.

12. 3−1

13. 5−2

14. 8−2

15. −5−2

16. −8−2

17. (−5)−2

18. (−8)−2

19.

20.

21.

22.

23.

Exercise: Simplifying expressions with negative exponents

PDF Transcript for Exercise: Simplifying expressions with negative exponents

24.

25. (10ab)0

26. (13x)0

27. 10ab0

28. 13x0

For Exercises 29–80, simplify and write the answer with positive exponents only. (See

Examples 2–5.)

29. y3 · y

5

30. x4 · x

8

31.

32.

33. (y2)

4

34. (z3)

4

35. (3x2)

4

Exercise: Simplifying an expression with exponents

PDF Transcript for Exercise: Simplifying an expression with exponents

36. (2y5)

3

37. p−3

38. q−5

39. 710 · 7

−13

40. 11−9

· 117

41.

42.

43. a−2a

−5

44. b−1b

−8

45.

46.

47.

48.

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49.

50.

51. (6xyz2)

0

52. (−7ab3)

0

53. 24 + 2

−2

54. 32 + 3

−1

55. 1−2 + 5

−2

56. 4−2 + 2

−2

57.

58.

59.

60.

61.

62.

63.

64.

65. (−3x−4

y5z

2)

−4

66. (−6a−2

b3c)

−2

67. (4m−2

n)(−m6n

−3)

68. (−6pq−3

)(2p4q)

69. (p−2q)

3 (2pq

4)

2

Exercise: Simplifying expressions with exponents

PDF Transcript for Exercise: Simplifying expressions with exponents

70. (mn3)

2 (5m

−2n

2)

71.

72.

73.

74.

75.

Exercise: Simplifying rational expression with exponents

PDF Transcript for Exercise: Simplifying rational expression with exponents

76.

77.

78.

79.

80.

Section 5.2 Practice Exercises

Vocabulary and Key Concepts

1.

1. A in the variable, x, is a single term or a sum of terms of the form axn, where a is

a real number and n is a nonnegative integer.

2. Given the term axn, a is called the , and is called the degree of the term.

3. Given the term x, the coefficient of the term is and the degree is .

4. A monomial is a polynomial with exactly term(s).

5. A is a polynomial with exactly two terms.

6. A is a polynomial with exactly three terms.

7. The term with the highest degree is called the term and its coefficient is called the

.

8. The degree of a polynomial is the degree of all of its terms.

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9. The degree of a nonzero constant such as 7 is .

10. If a term of a polynomial has more than one variable, then the degree of the term

is the sum of the of the variables contained in the term.

11. A function is a function defined by a finite sum of terms of the form axn, where a

is a real number and n is a whole number.

Review Exercises

For Exercises 2–6, simplify the expression.

2.

3. (2ac−2

)(5a−1

c4)

4.

5. (3.4 × 105)(5.0 × 10

−2)

6.

Concept 1: Polynomials: Basic Definitions

For Exercises 7–12, write the polynomial in descending order. Then identify the leading

coefficient and the degree.

7. a2 − 6a

3 − a

8. 2b − b4 + 5b

2

9. 6x2 − x + 3x

4 − 1

10. 8 − 4y + y5 − y

2

Exercise: Polynomials: Descending Order, Leading Coefficent, and Degree

PDF Transcript for Exercise: Polynomials: Descending Order, Leading Coefficent, and

Degree

11. 100 − t2

12. −51 + s2

For Exercises 13–18, write a polynomial in one variable that is described by the

following. (Answers may vary.)

13. A monomial of degree 5

14. A monomial of degree 4

15. A trinomial of degree 2

16. A trinomial of degree 3

17. A binomial of degree 4

18. A binomial of degree 2

Concept 2: Addition of Polynomials

For Exercises 19–30, add the polynomials and simplify. (See Examples 1 and 2.)

19. (−4m2 + 4m) + (5m

2 + 6m)

20. (3n3 + 5n) + (2n

3 − 2n)

21. (3x4 − x

3 − x

2) + (3x

3 − 7x

2 + 2x)

22. (6x3 − 2x

2 − 12) + (x

2 + 3x + 9)

Exercise: Adding Polynomials

PDF Transcript for Exercise: Adding Polynomials

23.

24.

25. Add (9x2y − 5xy + 1) to (8x

2y + xy − 15).

26. Add (−x3y

2 + 5xy) to (10x

3y

2 + x

2y − 10).

27. Add (−7a + 6a2 + 1) to (−8 − 4a − 2a

2).

28. Add (1 − 12p + 8p3) to (6p

2 + p

3 − 14).

29.

30.

Concept 3: Subtraction of Polynomials

For Exercises 31–36, write the opposite of the given polynomial. (See Example 3.)

31. −30y3

32. −2x2

33. 4p3 + 2p − 12

34. 8t2 − 4t − 3

35. −11ab2 + a

2b

36. −23rs − 4r + 9s

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For Exercises 37–46, subtract the polynomials and simplify. (See Examples 4 and 5.)

37. (13z5 − z

2) − (7z

5 + 5z

2)

38. (8w4 + 3w

2) − (12w

4 − w

2)

39. (−3x3 + 3x

2 − x + 6) − (1 − x − x

2 − x

3)

40. (−8x3 + 6x + 7) − (−4 − 2x − 5x

3)

41. (−3xy3 + 3x

2y − x + 6) − (−xy

3 − xy − x + 1)

42. (−8x2y

2 + 6xy

2 + 7xy) − (5xy

2 − 2xy − 4)

43.

44.

45.

Exercise: Subtracting Polynomials

PDF Transcript for Exercise: Subtracting Polynomials

46.

47. Subtract (9x2 − 5x + 1) from (8x

2 + x − 15). (See Example 6.)

48. Subtract (−x3 + 5x) from (10x

3 + x

2 − 10).

49. Find the difference of (3x5 − 2x

3 + 4) and (x

4 + 2x

3 − 7).

50. Find the difference of (7x10

− 2x4 − 3x) and (−4x

3 − 5x

4 + x + 5).

Mixed Exercises

For Exercises 51–74, add or subtract as indicated. Write the answers in descending order, if

possible.

51. (8y2 − 4y

3) − (3y

2 − 8y

3)

52. (−9y2 − 8) − (4y

2 + 3)

53. (−2r − 6r4) + (−r

4 − 9r)

54. (−8s9 + 7s

2) + (7s

9 − s

2)

55. (5xy + 13x2 + 3y) − (4x

2 − 8y)

56. (6p2q − 2q) − (−2p

2q + 13)

57. (11ab − 23b2) + (7ab − 19b

2)

58. (−4x2y + 9) + (8x

2y − 12)

59. [2p − (3p + 5)] + (4p − 6) + 2

60. −(q − 2) − [4 − (2q − 3) + 5]

61. 5 − [2m2 − (4m

2 + 1)]

62. [4n3 − (n

3 + 4)] + 3n

3

63. (6x3 − 5) − (−3x

3 + 2x) − (2x

3 − 6x)

Exercise: Mixed Exercise: Subtracting 3 Polynomials

PDF Transcript for Exercise: Mixed PDF Transcript for Exercise: Subtracting 3

Polynomials

64. (9p4 − 2) + (7p

4 + 1) − (8p

4 − 10)

65. (−ab + 5a2b) − [7ab

2 − 2ab − (7a

2b + 2ab

2)]

66. (m3n

2 + 4m

2n) − [−5m

3n

2 − 4mn − (7m

2n − 6mn)]

67. (8x3 − x

2 + 3) − [5x

2 + x − (4x

3 + x − 2)]

68. (y2 + 6y − 6) − [(2y

3 − 4y) − (3y

2 + y + 1)]

69.

Exercise: Mixed Exercise: Vertical Subtraction

PDF Transcript for Exercise: Mixed PDF Transcript for Exercise: Vertical Subtraction

70.

71.

72.

73.

74.

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For Exercises 75 and 76, find the perimeter.

75.

76.

Concept 4: Polynomial Functions

For Exercises 77–84, determine whether the given function is a polynomial function. If it is a

polynomial function, state the degree. If not, state the reason why.

77.

78. k(x) = −7x4 − 0.3x + x

3

79.

80. q(x) = x2 − 4x

−3

81. g(x) = −7

82. g(x) = 4x

83. M(x) = |x| + 5x

84. N(x) = x2 + |x|

85. Given P(x) = −x4 + 2x − 5, find the function values. (See Example 7.)

1. P(2)

2. P(−1)

3. P(0)

4. P(1)

86. Given N(x) = −x2 + 5x, find the function values.

1. N(1)

2. N(−1)

3. N(2)

4. N(0)

87. Given find the function values.

1. H(0)

2. H(2)

3. H(−2)

4. H(−1)

88. Given find the function values.

1. K(0)

2. K(3)

3. K(−3)

4. K(−1)

89. A rectangular garden is designed to be 3 ft longer than it is wide. Let x represent the

width of the garden. Find a function P that represents the perimeter in terms of x. (See

Example 8.)

Exercise: Polynomial Functions

PDF Transcript for Exercise: Polynomial Functions

90. Pauline measures a rectangular conference room and finds that the length is 4 yd greater

than twice the width. Let x represent the width. Find a function P that represents the

perimeter in terms of x.

91. The cost in dollars of producing x calendars is C(x) = 5.40x + 99. The revenue for selling

x calendars is R(x) = 12x. To calculate profit, subtract the cost from the revenue.

1. Write and simplify a function P that represents profit in terms of x.

2. Find the profit of producing and selling 50 calendars.

92. The cost in dollars of producing x lawn chairs is C(x) = 4.5x + 10.1. The revenue for

selling x chairs is R(x) = 12.99x. To calculate profit, subtract the cost from the revenue.

1. Write and simplify a function P that represents profit in terms of x.

2. Find the profit of producing and selling 100 lawn chairs.

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93. The function defined by D(x) = 5.2x2 + 40.4x + 1636 approximates the average yearly

dormitory charge for 4-yr universities x years since 1990. D(x) is the cost in dollars, and x

represents the number of years since 1990. (See Example 9.)

1. Evaluate D(0) and D(18) and interpret their meaning in the context of this

problem.

2. If this trend continues, what will the annual dormitory charge be in the year 2015?

Source: U.S. National Center for Education Statistics

94. The population of bacteria in a culture can be modeled by P(t) = −0.01t3 + 12.96t + 10,

where t is the time in hours after the culture was started and P(t) is the population in

thousands.

1. Evaluate P(0) and P(14) and interpret their meaning in the context of this

problem.

2. Predict the population of bacteria 24 hr after the culture was started.

95. The number of women, W, to be paid child support in the United States can be

approximated by

where t is the number of years since 2000, and W(t) is the yearly total measured in

thousands. (Source: U.S. Bureau of the Census)

1. Evaluate W(0), W(5), and W(10).

2. Interpret the meaning of the function value W(10).

96. The total yearly amount of child support due (in billions of dollars) in the United States

can be approximated by

where t is the number of years since 2000, and D(t) is the amount due (in billions of

dollars).

1. Evaluate D(0), D(4), and D(8).

2. Interpret the meaning of the function value of D(8).

Expanding Your Skills

97. A toy rocket is shot from ground level at an angle of 60° from the horizontal. See the

figure. The x- and y-positions of the rocket (measured in feet) vary with time t according

to

1. Evaluate x(0) and y(0), and write the values as an ordered pair. Interpret the

meaning of these function values in the context of this problem. Match the

ordered pair with a point on the graph.

2. Evaluate x(1) and y(1) and write the values as an ordered pair. Interpret the

meaning of these function values in the context of this problem. Match the

ordered pair with a point on the graph.

3. Evaluate x(2) and y(2), and write the values as an ordered pair. Match the ordered

pair with a point on the graph.