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This is “Review Exercises and Sample Exam”, section 6.8 from the book Beginning Algebra (v. 1.0). For details on it (including licensing), click here.
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6.8 Review Exercises and Sample Exam
Review Exercises
Introduction to Factoring
Determine the missing factor.
1. 12x3−24x2+4x=4x( ? )
2. 10y4−35y3−5y2=5y2( ? )
3. −18a5+9a4−27a3=−9a3( ? )
4. −21x2y+7xy2−49xy=−7xy( ? )
Factor out the GCF.
5. 22x2+11x
6. 15y4−5y3
7. 18a3−12a2+30a
8. 12a5+20a3−4a
9. 9x3y2−18x2y2+27xy2
10. 16a5b5c−8a3b6+24a3b2c
Factor by grouping.
11. x2+2x−5x−10
12. 2x2−2x−3x+3
13. x3+5x2−3x−15
14. x3−6x2+x−6
15. x3−x2y−2x+2y
16. a2b2−2a3+6ab−3b3
Factoring Trinomials of the Form x2 + bx + c
Are the following factored correctly? Check by multiplying.
17. x2+5x+6=(x+6)(x−1)
18. x2+3x−10=(x+5)(x−2)
19. x2+6x+9=(x+3)2
20. x2−6x−9=(x−3)(x+3)
Factor.
21. x2−13x−14
22. x2+13x+12
23. y2+10y+25
24. y2−20y+100
25. a2−8a−48
26. b2−18b+45
27. x2+2x+24
28. x2−10x−16
29. a2+ab−2b2
30. a2b2+5ab−50
Factoring Trinomials of the Form ax2 + bx + c
Factor.
31. 5x2−27x−18
32. 3x2−14x+8
33. 4x2−28x+49
34. 9x2+48x+64
35. 6x2−29x−9
36. 8x2+6x+9
37. 60x2−65x+15
38. 16x2−40x+16
39. 6x3−10x2y+4xy2
40. 10x3y−82x2y2+16xy3
41. −y2+9y+36
42. −a2−7a+98
43. 16+142x−18x2
44. 45−132x−60x2
Factoring Special Binomials
Factor completely.
45. x2−81
46. 25x2−36
47. 4x2−49
48. 81x2−1
49. x2−64y2
50. 100x2y2−1
51. 16x4−y4
52. x4−81y4
53. 8x3−125
54. 27+y3
55. 54x4y−2xy4
56. 3x4y2+24xy5
57. 64x6−y6
58. x6+1
General Guidelines for Factoring Polynomials
Factor completely.
59. 8x3−4x2+20x
60. 50a4b4c+5a3b5c2
61. x3−12x2−x+12
62. a3−2a2−3ab+6b
63. −y2−15y+16
64. x2−18x+72
65. 144x2−25
66. 3x4−48
67. 20x2−41x−9
68. 24x2+14x−20
69. a4b−343ab4
70. 32x7y2+4xy8
Solving Equations by Factoring
Solve.
71. (x−9)(x+10)=0
72. −3x(x+8)=0
73. 6(x+1)(x−1)=0
74. (x−12)(x+4)(2x−1)=0
75. x2+5x−50=0
76. 3x2−13x+4=0
77. 3x2−12=0
78. 16x2−9=0
79. (x−2)(x+6)=20
80. 2(x−2)(x+3)=7x−9
81. 52x2−203x=0
82. 23x2−512x+124=0
Find a quadratic equation with integer coefficients, given the following solutions.
83. −7, 6
84. 0, −10
85. −1/9, 1/2
86. ±3/2
Applications Involving Quadratic Equations
Set up an algebraic equation and then solve the following.
87. An integer is 4 less than twice another. If the product of the two integers is 96, then find the integers.
88. The sum of the squares of two consecutive positive even integers is 52. Find the integers.
89. A 20-
90. The height of an object dropped from the top of a 196-
91. The length of a rectangle is 1 centimeter less than three times the width. If the area is 70 square centimeters, then find the dimensions of the rectangle.
92. The base of a triangle is 4 centimeters more than twice the height. If the area of the triangle is 80 square centimeters, then find the measure of the base.
Sample Exam
1. Determine the GCF of the terms 25a2b2c, 50ab4, and 35a3b3c2.
2. Determine the missing factor: 24x2y3−16x3y2+8x2y=8x2y( ? ).
Factor.
3. 12x5−15x4+3x2
4. x3−4x2−2x+8
5. x2−7x+12
6. 9x2−12x+4
7. x2−81
8. x3+27y3
Factor completely.
9. x3+2x2−4x−8
10. x4−1
11. −6x3+20x2−6x
12. x6−1
Solve.
13. (2x+1)(x−7)=0
14. 3x(4x−3)(x+1)=0
15. x2−64=0
16. x2+4x−12=0
17. 23x2+89x−16=0
18. (x−5)(x−3)=−1
19. 3x(x+3)=14x+2
20. (3x+1)(3x+2)=9x+3
For each problem, set up an algebraic equation and then solve.
21. An integer is 4 less than twice another. If the product of the two integers is 70, then find the integers.
22. The sum of the squares of two consecutive positive odd integers is 130. Find the integers.
23. The length of a rectangle is 4 feet more than twice its width. If the area is 160 square feet, then find the dimensions of the rectangle.
24. The height of a triangle is 6 centimeters less than four times the length of its base. If the area measures 27 square centimeters, then what is the height of the triangle?
25. The height of a projectile launched upward at a speed of 64 feet/second from a height of 36 feet is given by the function h(t)=−16t2+64t+36. How long will it take the projectile to hit the ground?
Review Exercises Answers
1: (3x2−6x+1)
3: (2a2−a+3)
5: 11x(2x+1)
7: 6a(3a2−2a+5)
9: 9xy2(x2−2x+3)
11: (x+2)(x−5)
13: (x+5)(x2−3)
15: (x−y)(x2−2)
17: No
19: Yes
21: (x−14)(x+1)
23: (y+5)2
25: (a−12)(a+4)
27: Prime
29: (a−b)(a+2b)
31: (5x+3)(x−6)
33: (2x−7)2
35: Prime
37: 5(3x−1)(4x−3)
39: 2x(3x−2y)(x−y)
41: −1(y−12)(y+3)
43: −2(9x+1)(x−8)
45: (x+9)(x−9)
47: (2x+7)(2x−7)
49: (x+8y)(x−8y)
51: (4x2+y2)(2x+y)(2x−y)
53: (2x−5)(4x2+10x+25)
55: 2xy(3x−y)(9x2+3xy+y2)
57: (2x+y)(4x2−2xy+y2)(2x−y)(4x2+2xy+y2)
59: 4x(2x2−x+5)
61: (x−12)(x+1)(x−1)
63: −1(y+16)(y−1)
65: (12x+5)(12x−5)
67: (4x−9)(5x+1)
69: ab(a−7b)(a2+7ab+49b2)
71: 9, −10
73: −1, 1
75: −10, 5
77: ±2
79: −8, 4
81: 0, 8/3
83: x2+x−42=0
85: 18x2−7x−1=0
87: 8, 12 or −6, −16
89: 16 feet
91: Length: 14 centimeters; width: 5 centimeters
Sample Exam Answers
1: 5ab2
3: 3x2(4x3−5x2+1)
5: (x−4)(x−3)
7: (x+9)(x−9)
9: (x+2)2(x−2)
11: −2x(3x−1)(x−3)
13: −1/2, 7
15: ±8
17: −3/2, 1/6
19: −1/3, 2
21: 7, 10 or −14, −5
23: Width: 8 feet; length: 20 feet
25: 4½ sec
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