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This is “Review Exercises and Sample Exam”, section 5.7 from the book Beginning Algebra (v. 1.0). For details on it (including licensing), click here.
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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.
Adding and Subtracting Polynomials
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-
a. Find a formula for the profit. (profit = revenue − cost)
b. Use the formula to calculate the profit from producing and selling 150 t-
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
Sample Exam Answers
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
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