8.3E: Exercises
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Practice Makes Perfect
Factor Trinomials of the Form \(x^2+bx+c\)
In the following exercises, factor each trinomial of the form \(x^2+bx+c\).
1. \(p^2+11p+30\)
- Answer
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\((p+5)(p+6)\)
2. \(w^2+10w+21\)
3. \(n^2+19n+48\)
- Answer
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\((n+3)(n+16)\)
4. \(b^2+14b+48\)
5. \(a^2+25a+100\)
- Answer
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\((a+5)(a+20)\)
6. \(u^2+101u+100\)
7. \(x^2−8x+12\)
- Answer
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\((x−2)(x−6)\)
8. \(q^2−13q+36\)
9. \(y^2−18y+45\)
- Answer
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\((y−3)(y−15)\)
10. \(m^2−13m+30\)
11. \(x^2−8x+7\)
- Answer
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\((x−1)(x−7)\)
12. \(y^2−5y+6\)
13. \(5p−6+p^2\)
- Answer
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\((p−1)(p+6)\)
14. \(6n−7+n^2\)
15. \(8−6x+x^2\)
- Answer
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\((x−4)(x−2)\)
16. \(7x+x^2+6\)
17. \(x^2−12−11x\)
- Answer
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\((x−12)(x+1)\)
18. \(−11−10x+x^2\)
In the following exercises, factor each trinomial of the form \(x^2+bxy+cy^2\).
19. \(x^2−2xy−80y^2\)
- Answer
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\((x+8y)(x−10y)\)
20. \(p^2−8pq−65q^2\)
21. \(m^2−64mn−65n^2\)
- Answer
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\((m+n)(m−65n)\)
22. \(p^2−2pq−35q^2\)
23. \(a^2+5ab−24b^2\)
- Answer
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\((a+8b)(a−3b)\)
24. \(r^2+3rs−28s^2\)
25. \(x^2−3xy−14y^2\)
- Answer
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Prime
26. \(u^2−8uv−24v^2\)
27. \(m^2−5mn+30n^2\)
- Answer
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Prime
28. \(c^2−7cd+18d^2\)
Factor Trinomials of the Form \(ax^2+bx+c\) Using Trial and Error
In the following exercises, factor completely using trial and error.
29. \(p^3−8p^2−20p\)
- Answer
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\(p(p−10)(p+2)\)
30. \(q^3−5q^2−24q\)
31. \(3m^3−21m^2+30m\)
- Answer
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\(3m(m−5)(m−2)\)
32. \(11n^3−55n^2+44n\)
33. \(5x^4+10x^3−75x^2\)
- Answer
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\(5x^2(x−3)(x+5)\)
34. \(6y^4+12y^3−48y^2\)
35. \(2t^2+7t+5\)
- Answer
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\((2t+5)(t+1)\)
36. \(5y^2+16y+11\)
37. \(11x^2+34x+3\)
- Answer
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\((11x+1)(x+3)\)
38. \(7b^2+50b+7\)
39. \(4w^2−5w+1\)
- Answer
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\((4w−1)(w−1)\)
40. \(5x^2−17x+6\)
41. \(4q^2−7q−2\)
- Answer
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\((4q+1)(q−2)\)
42. \(10y^2−53y−111\)
43. \(6p^2−19pq+10q^2\)
- Answer
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\((2p−5q)(3p−2q)\)
44. \(21m^2−29mn+10n^2\)
45. \(4a^2+17ab−15b^2\)
- Answer
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\((4a−3b)(a+5b)\)
46. \(6u^2+5uv−14v^2\)
47. \(−16x^2−32x−16\)
- Answer
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\(−16(x+1)(x+1)\)
48. \(−81a^2+153a+18\)
49. \(−30q^3−140q^2−80q\)
- Answer
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\( - 10q(3q+2)(q+4)\)
50. \(−5y^3−30y^2+35y\)
Factor Trinomials of the Form \(ax^2+bx+c\) using the ‘ac’ Method
In the following exercises, factor using the ‘ac’ method.
51. \(5n^2+21n+4\)
- Answer
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\((5n+1)(n+4)\)
52. \(8w^2+25w+3\)
53. \(4k^2−16k+15\)
- Answer
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\((2k−3)(2k−5)\)
54. \(5s^2−9s+4\)
55. \(6y^2+y−15\)
- Answer
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\((3y+5)(2y−3)\)
56. \(6p^2+p−22\)
57. \(2n^2−27n−45\)
- Answer
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\((2n+3)(n−15)\)
58. \(12z^2−41z−11\)
59. \(60y^2+290y−50\)
- Answer
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\(10(6y−1)(y+5)\)
60. \(6u^2−46u−16\)
61. \(48z^3−102z^2−45z\)
- Answer
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\(3z(8z+3)(2z−5)\)
62. \(90n^3+42n^2−216n\)
63. \(16s^2+40s+24\)
- Answer
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\(8(2s+3)(s+1)\)
64. \(24p^2+160p+96\)
65. \(48y^2+12y−36\)
- Answer
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\(12(4y−3)(y+1)\)
66. \(30x^2+105x−60\)
Factor Using Substitution
In the following exercises, factor using substitution.
67. \(x^4−x^2−12\)
- Answer
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\((x^2+3)(x^2−4)\)
68. \(x^4+2x^2−8\)
69. \(x^4−3x^2−28\)
- Answer
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\((x^2−7)(x^2+4)\)
70. \(x^4−13x^2−30\)
71. \((x−3)^2−5(x−3)−36\)
- Answer
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\((x−12)(x+1)\)
72. \((x−2)^2−3(x−2)−54\)
73. \((3y−2)^2−(3y−2)−2\)
- Answer
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\((3y−4)(3y−1)\)
74. \((5y−1)^2−3(5y−1)−18\)
Mixed Practice
In the following exercises, factor each expression using any method.
75. \(u^2−12u+36\)
- Answer
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\((u−6)(u−6)\)
76. \(x^2−14x−32\)
77. \(r^2−20rs+64s^2\)
- Answer
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\((r−4s)(r−16s)\)
78. \(q^2−29qr−96r^2\)
79. \(12y^2−29y+14\)
- Answer
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\((4y−7)(3y−2)\)
80. \(12x^2+36y−24z\)
81. \(6n^2+5n−4\)
- Answer
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\((2n−1)(3n+4)\)
82. \(3q^2+6q+2\)
83. \(13z^2+39z−26\)
- Answer
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\(13(z^2+3z−2)\)
84. \(5r^2+25r+30\)
85. \(3p^2+21p\)
- Answer
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\(3p(p+7)\)
86. \(7x^2−21x\)
87. \(6r^2+30r+36\)
- Answer
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\(6(r+2)(r+3)\)
88. \(18m^2+15m+3\)
89. \(24n^2+20n+4\)
- Answer
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\(4(2n+1)(3n+1)\)
90. \(4a^2+5a+2\)
91. \(x^4−4x^2−12\)
- Answer
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\((x^2+2)(x^2−6)\)
92. \(x^4−7x^2−8\)
93. \((x+3)^2−9(x+3)−36\)
- Answer
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\((x−9)(x+6)\)
94. \((x+2)^2−25(x+2)−54\)
Writing Exercises
95. Many trinomials of the form \(x^2+bx+c\) factor into the product of two binomials \((x+m)(x+n)\). Explain how you find the values of \(m\) and \(n\).
- Answer
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Answers will vary.
96. Tommy factored \(x^2−x−20\) as \((x+5)(x−4)\). Sara factored it as \((x+4)(x−5)\). Ernesto factored it as \((x−5)(x−4)\). Who is correct? Explain why the other two are wrong.
97. List, in order, all the steps you take when using the “\(ac\)” method to factor a trinomial of the form \(ax^2+bx+c\).
- Answer
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Answers will vary.
98. How is the “\(ac\)” method similar to the “undo FOIL” method? How is it different?
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. After reviewing this checklist, what will you do to become confident for all objectives?