Review Exercises and Sample Exam

5.7 Review Exercises and Sample Exam

Review Exercises

Rules of Exponents

Simplify.

1. $7^{3} \cdot 7^{6}$

2. $\frac{5^{9}}{5^{6}}$

3. $y^{5} \cdot y^{2} \cdot y^{3}$

4. $x^{3} y^{2} \cdot x y^{3}$

5. $- 5 a^{3} b^{2} c \cdot 6 a^{2} b c^{2}$

6. $\frac{55 x^{2} y z^{5}}{5 x y z^{2}}$

7. ${(\frac{- 3 a^{2} b^{4}}{2 c^{3}})}^{2}$

8. ${(\frac{- 2 a^{3} b}{4 c^{4}})}^{3}$

9. $- 5 x^{3} y^{0} {(z^{2})}^{3} \cdot 2 x^{4} {(y^{3})}^{2} z$

10. ${(- 25 x^{6} y^{5} z)}^{0}$

11. Each side of a square measures $5 x^{2}$ units. Find the area of the square in terms of x.

12. Each side of a cube measures $2 x^{3}$ units. Find the volume of the cube in terms of x.

Introduction to Polynomials

Classify the given polynomial as a monomial, binomial, or trinomial and state the degree.

13. $8 a^{3} - 1$

14. $5 y^{2} - y + 1$

15. $- 12 a b^{2}$

16. 10

Write the following polynomials in standard form.

17. $7 - x^{2} - 5 x$

18. $5 x^{2} - 1 - 3 x + 2 x^{3}$

Evaluate.

19. $2 x^{2} - x + 1$ , where $x = - 3$

20. $\frac{1}{2} x - \frac{3}{4}$ , where $x = \frac{1}{3}$

21. $b^{2} - 4 a c$ , where $a = - \frac{1}{2}$ , $b = - 3$ , and $c = - \frac{3}{2}$

22. $a^{2} - b^{2}$ , where $a = - \frac{1}{2}$ and $b = - \frac{1}{3}$

23. $a^{3} - b^{3}$ , where $a = - 2$ and $b = - 1$

24. $x y^{2} - 2 x^{2} y$ , where $x = - 3$ and $y = - 1$

25. Given $f (x) = 3 x^{2} - 5 x + 2$ , find $f (- 2)$ .

26. Given $g (x) = x^{3} - x^{2} + x - 1$ , find $g (- 1)$ .

27. The surface area of a rectangular solid is given by the formula $S A = 2 l w + 2 w h + 2 l h$ , where l, w, and h represent the length, width, and height, respectively. If the length of a rectangular solid measures 2 units, the width measures 3 units, and the height measures 5 units, then calculate the surface area.

28. The surface area of a sphere is given by the formula $S A = 4 π r^{2}$ , where r represents the radius of the sphere. If a sphere has a radius of 5 units, then calculate the surface area.

Adding and Subtracting Polynomials

Perform the operations.

29. $(3 x - 4) + (9 x - 1)$

30. $(\frac{1}{3} x - \frac{1}{9}) + (\frac{1}{6} x + \frac{1}{2})$

31. $(7 x^{2} - x + 9) + (x^{2} - 5 x + 6)$

32. $(6 x^{2} y - 5 x y^{2} - 3) + (- 2 x^{2} y + 3 x y^{2} + 1)$

33. $(4 y + 7) - (6 y - 2) + (10 y - 1)$

34. $(5 y^{2} - 3 y + 1) - (8 y^{2} + 6 y - 11)$

35. $(7 x^{2} y^{2} - 3 x y + 6) - (6 x^{2} y^{2} + 2 x y - 1)$

36. $(a^{3} - b^{3}) - (a^{3} + 1) - (b^{3} - 1)$

37. $(x^{5} - x^{3} + x - 1) - (x^{4} - x^{2} + 5)$

38. $(5 x^{3} - 4 x^{2} + x - 3) - (5 x^{3} - 3) + (4 x^{2} - x)$

39. Subtract $2 x - 1$ from $9 x + 8$ .

40. Subtract $3 x^{2} - 10 x - 2$ from $5 x^{2} + x - 5$ .

41. Given $f (x) = 3 x^{2} - x + 5$ and $g (x) = x^{2} - 9$ , find $(f + g) (x)$ .

42. Given $f (x) = 3 x^{2} - x + 5$ and $g (x) = x^{2} - 9$ , find $(f - g) (x)$ .

43. Given $f (x) = 3 x^{2} - x + 5$ and $g (x) = x^{2} - 9$ , find $(f + g) (- 2)$ .

44. Given $f (x) = 3 x^{2} - x + 5$ and $g (x) = x^{2} - 9$ , find $(f - g) (- 2)$ .

Multiplying Polynomials

Multiply.

45. $6 x^{2} (- 5 x^{4})$

46. $3 a b^{2} (7 a^{2} b)$

47. $2 y (5 y - 12)$

48. $- 3 x (3 x^{2} - x + 2)$

49. $x^{2} y (2 x^{2} y - 5 x y^{2} + 2)$

50. $- 4 a b (a^{2} - 8 a b + b^{2})$

51. $(x - 8) (x + 5)$

52. $(2 y - 5) (2 y + 5)$

53. ${(3 x - 1)}^{2}$

54. ${(3 x - 1)}^{3}$

55. $(2 x - 1) (5 x^{2} - 3 x + 1)$

56. $(x^{2} + 3) (x^{3} - 2 x - 1)$

57. ${(5 y + 7)}^{2}$

58. ${(y^{2} - 1)}^{2}$

59. Find the product of $x^{2} - 1$ and $x^{2} + 1$ .

60. Find the product of $\frac{3}{2} x^{2} y$ and $10 x - 30 y + 2$ .

61. Given $f (x) = 7 x - 2$ and $g (x) = x^{2} - 3 x + 1$ , find $(f \cdot g) (x)$ .

62. Given $f (x) = x - 5$ and $g (x) = x^{2} - 9$ , find $(f \cdot g) (x)$ .

63. Given $f (x) = 7 x - 2$ and $g (x) = x^{2} - 3 x + 1$ , find $(f \cdot g) (- 1)$ .

64. Given $f (x) = x - 5$ and $g (x) = x^{2} - 9$ , find $(f \cdot g) (- 1)$ .

Dividing Polynomials

Divide.

65. $\frac{7 y^{2} - 14 y + 28}{7}$

66. $\frac{12 x^{5} - 30 x^{3} + 6 x}{6 x}$

67. $\frac{4 a^{2} b - 16 a b^{2} - 4 a b}{- 4 a b}$

68. $\frac{6 a^{6} - 24 a^{4} + 5 a^{2}}{3 a^{2}}$

69. $(10 x^{2} - 19 x + 6) \div (2 x - 3)$

70. $(2 x^{3} - 5 x^{2} + 5 x - 6) \div (x - 2)$

71. $\frac{10 x^{4} - 21 x^{3} - 16 x^{2} + 23 x - 20}{2 x - 5}$

72. $\frac{x^{5} - 3 x^{4} - 28 x^{3} + 61 x^{2} - 12 x + 36}{x - 6}$

73. $\frac{10 x^{3} - 55 x^{2} + 72 x - 4}{2 x - 7}$

74. $\frac{3 x^{4} + 19 x^{3} + 3 x^{2} - 16 x - 11}{3 x + 1}$

75. $\frac{5 x^{4} + 4 x^{3} - 5 x^{2} + 21 x + 21}{5 x + 4}$

76. $\frac{x^{4} - 4}{x - 4}$

77. $\frac{2 x^{4} + 10 x^{3} - 23 x^{2} - 15 x + 30}{2 x^{2} - 3}$

78. $\frac{7 x^{4} - 17 x^{3} + 17 x^{2} - 11 x + 2}{x^{2} - 2 x + 1}$

79. Given $f (x) = x^{3} - 4 x + 1$ and $g (x) = x - 1$ , find $(f / g) (x)$ .

80. Given $f (x) = x^{5} - 32$ and $g (x) = x - 2$ , find $(f / g) (x)$ .

81. Given $f (x) = x^{3} - 4 x + 1$ and $g (x) = x - 1$ , find $(f / g) (2)$ .

82. Given $f (x) = x^{5} - 32$ and $g (x) = x - 2$ , find $(f / g) (0)$ .

Negative Exponents

Simplify.

83. ${(- 10)}^{- 2}$

84. $- 10^{- 2}$

85. $5 x^{- 3}$

86. ${(5 x)}^{- 3}$

87. $\frac{1}{7 y^{- 3}}$

88. $\frac{3 x^{- 4}}{y^{- 2}}$

89. $\frac{- 2 a^{2} b^{- 5}}{c^{- 8}}$

90. ${(- 5 x^{2} y z^{- 1})}^{- 2}$

91. ${(- 2 x^{- 3} y^{0} z^{2})}^{- 3}$

92. ${(\frac{- 10 a^{5} b^{3} c^{2}}{5 a b^{2} c^{2}})}^{- 1}$

93. ${(\frac{a^{2} b^{- 4} c^{0}}{2 a^{4} b^{- 3} c})}^{- 3}$

The value in dollars of a new laptop computer can be estimated by using the formula $V = 1200 {(t + 1)}^{- 1}$ , where t represents the number of years after the purchase.

94. Estimate the value of the laptop when it is 1½ years old.

95. What was the laptop worth new?

Rewrite using scientific notation.

96. 2,030,000,000

97. 0.00000004011

Perform the indicated operations.

98. $(5.2 \times 10^{12}) (1.8 \times 10^{- 3})$

99. $(9.2 \times 10^{- 4}) (6.3 \times 10^{22})$

100. $\frac{4 \times 10^{16}}{8 \times 10^{- 7}}$

101. $\frac{9 \times 10^{- 30}}{4 \times 10^{- 10}}$

102. 5,000,000,000,000 × 0.0000023

103. 0.0003/120,000,000,000,000

Sample Exam

Simplify.

1. $- 5 x^{3} (2 x^{2} y)$

2. ${(x^{2})}^{4} \cdot x^{3} \cdot x$

3. $\frac{{(- 2 x^{2} y^{3})}^{2}}{x^{2} y}$

4. a. ${(- 5)}^{0}$ ; b. $- 5^{0}$

Evaluate.

5. $2 x^{2} - x + 5$ , where $x = - 5$

6. $a^{2} - b^{2}$ , where $a = 4$ and $b = - 3$

Perform the operations.

7. $(3 x^{2} - 4 x + 5) + (- 7 x^{2} + 9 x - 2)$

8. $(8 x^{2} - 5 x + 1) - (10 x^{2} + 2 x - 1)$

9. $(\frac{3}{5} a - \frac{1}{2}) - (\frac{2}{3} a^{2} + \frac{2}{3} a - \frac{2}{9}) + (\frac{1}{15} a - \frac{5}{18})$

10. $2 x^{2} (2 x^{3} - 3 x^{2} - 4 x + 5)$

11. $(2 x - 3) (x + 5)$

12. ${(x - 1)}^{3}$

13. $\frac{81 x^{5} y^{2} z}{- 3 x^{3} y z}$

14. $\frac{10 x^{9} - 15 x^{5} + 5 x^{2}}{- 5 x^{2}}$

15. $\frac{x^{3} - 5 x^{2} + 7 x - 2}{x - 2}$

16. $\frac{6 x^{4} - x^{3} - 13 x^{2} - 2 x - 1}{2 x - 1}$

Simplify.

17. $2^{- 3}$

18. $- 5 x^{- 2}$

19. ${(2 x^{4} y^{- 3} z)}^{- 2}$

20. ${(\frac{- 2 a^{3} b^{- 5} c^{- 2}}{a b^{- 3} c^{2}})}^{- 3}$

21. Subtract $5 x^{2} y - 4 x y^{2} + 1$ from $10 x^{2} y - 6 x y^{2} + 2$ .

22. If each side of a cube measures $4 x^{4}$ units, calculate the volume in terms of x.

23. The height of a projectile in feet is given by the formula $h = - 16 t^{2} + 96 t + 10$ , where t represents time in seconds. Calculate the height of the projectile at 1½ seconds.

24. The cost in dollars of producing custom t-shirts is given by the formula $C = 120 + 3.50 x$ , where x represents the number of t-shirts produced. The revenue generated by selling the t-shirts for $6.50 each is given by the formula $R = 6.50 x$ , where x represents the number of t-shirts sold.

a. Find a formula for the profit. (profit = revenue − cost)

b. Use the formula to calculate the profit from producing and selling 150 t-shirts.

25. The total volume of water in earth’s oceans, seas, and bays is estimated to be $4.73 \times 10^{19}$ cubic feet. By what factor is the volume of the moon, $7.76 \times 10^{20}$ cubic feet, larger than the volume of earth’s oceans? Round to the nearest tenth.

Review Exercises Answers

1: $7^{9}$

3: $y^{10}$

5: $- 30 a^{5} b^{3} c^{3}$

7: $\frac{9 a^{4} b^{8}}{4 c^{6}}$

9: $- 10 x^{7} y^{6} z^{7}$

11: $A = 25 x^{4}$

13: Binomial; degree 3

15: Monomial; degree 3

17: $- x^{2} - 5 x + 7$

19: 22

21: 6

23: −7

25: $f (- 2) = 24$

27: 62 square units

29: $12 x - 5$

31: $8 x^{2} - 6 x + 15$

33: $8 y + 8$

35: $x^{2} y^{2} - 5 x y + 7$

37: $x^{5} - x^{4} - x^{3} + x^{2} + x - 6$

39: $7 x + 9$

41: $(f + g) (x) = 4 x^{2} - x - 4$

43: $(f + g) (- 2) = 14$

45: $- 30 x^{6}$

47: $10 y^{2} - 24 y$

49: $2 x^{4} y^{2} - 5 x^{3} y^{3} + 2 x^{2} y$

51: $x^{2} - 3 x - 40$

53: $9 x^{2} - 6 x + 1$

55: $10 x^{3} - 11 x^{2} + 5 x - 1$

57: $25 y^{2} + 70 y + 49$

59: $x^{4} - 1$

61: $(f \cdot g) (x) = 7 x^{3} - 23 x^{2} + 13 x - 2$

63: $(f \cdot g) (- 1) = - 45$

65: $y^{2} - 2 y + 4$

67: $- a + 4 b + 1$

69: $5 x - 2$

71: $5 x^{3} + 2 x^{2} - 3 x + 4$

73: $5 x^{2} - 10 x + 1 + \frac{3}{2 x - 7}$

75: $x^{3} - x + 5 + \frac{1}{5 x + 4}$

77: $x^{2} + 5 x - 10$

79: $(f / g) (x) = x^{2} + x - 3 - \frac{2}{x - 1}$

81: $(f / g) (2) = 1$

83: $\frac{1}{100}$

85: $\frac{5}{x^{3}}$

87: $\frac{y^{3}}{7}$

89: $\frac{- 2 a^{2} c^{8}}{b^{5}}$

91: $- \frac{x^{9}}{8 z^{6}}$

93: $8 a^{6} b^{3} c^{3}$

95: $1,200

97: $4.011 \times 10^{- 8}$

99: $5.796 \times 10^{19}$

101: $2.25 \times 10^{- 20}$

103: $2.5 \times 10^{- 18}$

Sample Exam Answers

1: $- 10 x^{5} y$

3: $4 x^{2} y^{5}$

5: 60

7: $- 4 x^{2} + 5 x + 3$

9: $- \frac{2}{3} a^{2} - \frac{5}{9}$

11: $2 x^{2} + 7 x - 15$

13: $- 27 x^{2} y$

15: $x^{2} - 3 x + 1$

17: $\frac{1}{8}$

19: $\frac{y^{6}}{4 x^{8} z^{2}}$

21: $5 x^{2} y - 2 x y^{2} + 1$

23: 118 feet

25: 16.4