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5.7 Review Exercises and Sample Exam

Review Exercises

Rules of Exponents

Simplify.

1. $73⋅76$

2. $5956$

3. $y5⋅y2⋅y3$

4. $x3y2⋅xy3$

5. $−5a3b2c⋅6a2bc2$

6. $55x2yz55xyz2$

7. $(−3 a 2 b 42 c 3)2$

8. $(−2 a 3b4 c 4)3$

9. $−5x3y0(z2)3⋅2x4(y3)2z$

10. $(−25x6y5z)0$

11. Each side of a square measures $5x2$ units. Find the area of the square in terms of x.

12. Each side of a cube measures $2x3$ 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. $8a3−1$

14. $5y2−y+1$

15. $−12ab2$

16. 10

Write the following polynomials in standard form.

17. $7−x2−5x$

18. $5x2−1−3x+2x3$

Evaluate.

19. $2x2−x+1$, where $x=−3$

20. $12x−34$, where $x=13$

21. $b2−4ac$, where $a=−12$, $b=−3$, and $c=−32$

22. $a2−b2$, where $a=−12$ and $b=−13$

23. $a3−b3$, where $a=−2$ and $b=−1$

24. $xy2−2x2y$, where $x=−3$ and $y=−1$

25. Given $f(x)=3x2−5x+2$, find $f(−2)$.

26. Given $g(x)=x3−x2+x−1$, find $g(−1)$.

27. The surface area of a rectangular solid is given by the formula $SA=2lw+2wh+2lh$, 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 $SA=4πr2$, where r represents the radius of the sphere. If a sphere has a radius of 5 units, then calculate the surface area.

Perform the operations.

29. $(3x−4)+(9x−1)$

30. $(13x−19)+(16x+12)$

31. $(7x2−x+9)+(x2−5x+6)$

32. $(6x2y−5xy2−3)+(−2x2y+3xy2+1)$

33. $(4y+7)−(6y−2)+(10y−1)$

34. $(5y2−3y+1)−(8y2+6y−11)$

35. $(7x2y2−3xy+6)−(6x2y2+2xy−1)$

36. $(a3−b3)−(a3+1)−(b3−1)$

37. $(x5−x3+x−1)−(x4−x2+5)$

38. $(5x3−4x2+x−3)−(5x3−3)+(4x2−x)$

39. Subtract $2x−1$ from $9x+8$.

40. Subtract $3x2−10x−2$ from $5x2+x−5$.

41. Given $f(x)=3x2−x+5$ and $g(x)=x2−9$, find $(f+g)(x)$.

42. Given $f(x)=3x2−x+5$ and $g(x)=x2−9$, find $(f−g)(x)$.

43. Given $f(x)=3x2−x+5$ and $g(x)=x2−9$, find $(f+g)(−2)$.

44. Given $f(x)=3x2−x+5$ and $g(x)=x2−9$, find $(f−g)(−2)$.

Multiplying Polynomials

Multiply.

45. $6x2(−5x4)$

46. $3ab2(7a2b)$

47. $2y(5y−12)$

48. $−3x(3x2−x+2)$

49. $x2y(2x2y−5xy2+2)$

50. $−4ab(a2−8ab+b2)$

51. $(x−8)(x+5)$

52. $(2y−5)(2y+5)$

53. $(3x−1)2$

54. $(3x−1)3$

55. $(2x−1)(5x2−3x+1)$

56. $(x2+3)(x3−2x−1)$

57. $(5y+7)2$

58. $(y2−1)2$

59. Find the product of $x2−1$ and $x2+1$.

60. Find the product of $32x2y$ and $10x−30y+2$.

61. Given $f(x)=7x−2$ and $g(x)=x2−3x+1$, find $(f⋅g)(x)$.

62. Given $f(x)=x−5$ and $g(x)=x2−9$, find $(f⋅g)(x)$.

63. Given $f(x)=7x−2$ and $g(x)=x2−3x+1$, find $(f⋅g)(−1)$.

64. Given $f(x)=x−5$ and $g(x)=x2−9$, find $(f⋅g)(−1)$.

Dividing Polynomials

Divide.

65. $7y2−14y+287$

66. $12x5−30x3+6x6x$

67. $4a2b−16ab2−4ab−4ab$

68. $6a6−24a4+5a23a2$

69. $(10x2−19x+6)÷(2x−3)$

70. $(2x3−5x2+5x−6)÷(x−2)$

71. $10x4−21x3−16x2+23x−202x−5$

72. $x5−3x4−28x3+61x2−12x+36x−6$

73. $10x3−55x2+72x−42x−7$

74. $3x4+19x3+3x2−16x−113x+1$

75. $5x4+4x3−5x2+21x+215x+4$

76. $x4−4x−4$

77. $2x4+10x3−23x2−15x+302x2−3$

78. $7x4−17x3+17x2−11x+2x2−2x+1$

79. Given $f(x)=x3−4x+1$ and $g(x)=x−1$, find $(f/g)(x)$.

80. Given $f(x)=x5−32$ and $g(x)=x−2$, find $(f/g)(x)$.

81. Given $f(x)=x3−4x+1$ and $g(x)=x−1$, find $(f/g)(2)$.

82. Given $f(x)=x5−32$ and $g(x)=x−2$, find $(f/g)(0)$.

Negative Exponents

Simplify.

83. $(−10)−2$

84. $−10−2$

85. $5x−3$

86. $(5x)−3$

87. $17y−3$

88. $3x−4y−2$

89. $−2a2b−5c−8$

90. $(−5x2yz−1)−2$

91. $(−2x−3y0z2)−3$

92. $(−10 a 5 b 3 c 25a b 2 c 2)−1$

93. $( a 2 b −4 c 02 a 4 b −3c)−3$

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. $(5.2×1012)(1.8×10−3)$

99. $(9.2×10−4)(6.3×1022)$

100. $4×10168×10−7$

101. $9×10−304×10−10$

102. 5,000,000,000,000 × 0.0000023

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

Sample Exam

Simplify.

1. $−5x3(2x2y)$

2. $(x2)4⋅x3⋅x$

3. $(−2 x 2 y 3)2x2y$

4. a. $(−5)0$; b. $−50$

Evaluate.

5. $2x2−x+5$, where $x=−5$

6. $a2−b2$, where $a=4$ and $b=−3$

Perform the operations.

7. $(3x2−4x+5)+(−7x2+9x−2)$

8. $(8x2−5x+1)−(10x2+2x−1)$

9. $(35a−12)−(23a2+23a−29)+(115a−518)$

10. $2x2(2x3−3x2−4x+5)$

11. $(2x−3)(x+5)$

12. $(x−1)3$

13. $81x5y2z−3x3yz$

14. $10x9−15x5+5x2−5x2$

15. $x3−5x2+7x−2x−2$

16. $6x4−x3−13x2−2x−12x−1$

Simplify.

17. $2−3$

18. $−5x−2$

19. $(2x4y−3z)−2$

20. $(−2 a 3 b −5 c −2a b −3 c 2)−3$

21. Subtract $5x2y−4xy2+1$ from $10x2y−6xy2+2$.

22. If each side of a cube measures $4x4$ units, calculate the volume in terms of x.

23. The height of a projectile in feet is given by the formula $h=−16t2+96t+10$, 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 $C=120+3.50x$, 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 $R=6.50x$, where x represents the number of t-shirts sold. a. Find a formula for the profit. (profit = revenuecost) 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 $4.73×1019$ cubic feet. By what factor is the volume of the moon, $7.76×1020$ cubic feet, larger than the volume of earth’s oceans? Round to the nearest tenth. Review Exercises Answers 1: $79$ 3: $y10$ 5: $−30a5b3c3$ 7: $9a4b84c6$ 9: $−10x7y6z7$ 11: $A=25x4$ 13: Binomial; degree 3 15: Monomial; degree 3 17: $−x2−5x+7$ 19: 22 21: 6 23: −7 25: $f(−2)=24$ 27: 62 square units 29: $12x−5$ 31: $8x2−6x+15$ 33: $8y+8$ 35: $x2y2−5xy+7$ 37: $x5−x4−x3+x2+x−6$ 39: $7x+9$ 41: $(f+g)(x)=4x2−x−4$ 43: $(f+g)(−2)=14$ 45: $−30x6$ 47: $10y2−24y$ 49: $2x4y2−5x3y3+2x2y$ 51: $x2−3x−40$ 53: $9x2−6x+1$ 55: $10x3−11x2+5x−1$ 57: $25y2+70y+49$ 59: $x4−1$ 61: $(f⋅g)(x)=7x3−23x2+13x−2$ 63: $(f⋅g)(−1)=−45$ 65: $y2−2y+4$ 67: $−a+4b+1$ 69: $5x−2$ 71: $5x3+2x2−3x+4$ 73: $5x2−10x+1+32x−7$ 75: $x3−x+5+15x+4$ 77: $x2+5x−10$ 79: $(f/g)(x)=x2+x−3−2x−1$ 81: $(f/g)(2)=1$ 83: $1100$ 85: $5x3$ 87: $y37$ 89: $−2a2c8b5$ 91: $−x98z6$ 93: $8a6b3c3$ 95:$1,200

97: $4.011×10−8$

99: $5.796×1019$

101: $2.25×10−20$

103: $2.5×10−18$

1: $−10x5y$

3: $4x2y5$

5: 60

7: $−4x2+5x+3$

9: $−23a2−59$

11: $2x2+7x−15$

13: $−27x2y$

15: $x2−3x+1$

17: $18$

19: $y64x8z2$

21: $5x2y−2xy2+1$

23: 118 feet

25: 16.4