14.4E: Exercises
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Practice Makes Perfect
Factor Perfect Square Trinomials
In the following exercises, factor completely using the perfect square trinomials pattern.
1. \(16y^2+24y+9\)
- Answer
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\((4y+3)^2\)
2. \(25v^2+20v+4\)
3. \(36s^2+84s+49\)
- Answer
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\((6s+7)^2\)
4. \(49s^2+154s+121\)
5. \(100x^2−20x+1\)
- Answer
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\((10x−1)^2\)
6. \(64z^2−16z+1\)
7. \(25n^2−120n+144\)
- Answer
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\((5n−12)^2\)
8. \(4p^2−52p+169\)
9. \(49x^2+28xy+4y^2\)
- Answer
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\((7x+2y)^2\)
10. \(25r^2+60rs+36s^2\)
11. \(100y^2−52y+1\)
- Answer
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\((50y−1)(2y−1)\)
12. \(64m^2−34m+1\)
13. \(10jk^2+80jk+160j\)
- Answer
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\(10j(k+4)^2\)
14. \(64x^2y−96xy+36y\)
15. \(75u^4−30u^3v+3u^2v^2\)
- Answer
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\(3u^2(5u−v)^2\)
16. \(90p^4+300p^4q+250p^2q^2\)
Factor Differences of Squares
In the following exercises, factor completely using the difference of squares pattern, if possible.
17. \(25v^2−1\)
- Answer
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\((5v−1)(5v+1)\)
18. \(169q^2−1\)
19. \(4−49x^2\)
- Answer
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\((7x−2)(7x+2)\)
20. \(121−25s^2\)
21. \(6p^2q^2−54p^2\)
- Answer
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\(6p^2(q−3)(q+3)\)
22. \(98r^3−72r\)
23. \(24p^2+54\)
- Answer
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\(6(4p^2+9)\)
24. \(20b^2+140\)
25. \(121x^2−144y^2\)
- Answer
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\((11x−12y)(11x+12y)\)
26. \(49x^2−81y^2\)
27. \(169c^2−36d^2\)
- Answer
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\((13c−6d)(13c+6d)\)
28. \(36p^2−49q^2\)
29. \(16z^4−1\)
- Answer
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\((2z−1)(2z+1)(4z^2+1)\)
30. \(m^4−n^4\)
31. \(162a^4b^2−32b^2\)
- Answer
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\(2b^2(3a−2)(3a+2)(9a^2+4)\)
32. \(48m^4n^2−243n^2\)
33. \(x^2−16x+64−y^2\)
- Answer
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\((x−8−y)(x−8+y)\)
34. \(p^2+14p+49−q^2\)
35. \(a^2+6a+9−9b^2\)
- Answer
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\((a+3−3b)(a+3+3b)\)
36. \(m^2−6m+9−16n^2\)
Factor Sums and Differences of Cubes
In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
37. \(x^3+125\)
- Answer
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\((x+5)(x^2−5x+25)\)
38. \(n^6+512\)
39. \(z^6−27\)
- Answer
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\((z^2−3)(z^4+3z^2+9)\)
40. \(v^3−216\)
41. \(8−343t^3\)
- Answer
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\((2−7t)(4+14t+49t^2)\)
42. \(125−27w^3\)
43. \(8y^3−125z^3\)
- Answer
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\((2y−5z)(4y^2+10yz+25z^2)\)
44. \(27x^3−64y^3\)
45. \(216a^3+125b^3\)
- Answer
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\((6a+5b)(36a^2−30ab+25b^2)\)
46. \(27y^3+8z^3\)
47. \(7k^3+56\)
- Answer
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\(7(k+2)(k^2−2k+4)\)
48. \(6x^3−48y^3\)
49. \(2x^2−16x^2y^3\)
- Answer
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\(2x^2(1−2y)(1+2y+4y^2)\)
50. \(−2x^3y^2−16y^5\)
51. \((x+3)^3+8x^3\)
- Answer
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\(9(x+1)(x^2+3)\)
52. \((x+4)^3−27x^3\)
53. \((y−5)^3−64y^3\)
- Answer
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\(−(3y+5)(21y^2−30y+25)\)
54. \((y−5)^3+125y^3\)
Mixed Practice
In the following exercises, factor completely.
55. \(64a^2−25\)
- Answer
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\((8a−5)(8a+5)\)
56. \(121x^2−144\)
57. \(27q^2−3\)
- Answer
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\(3(3q−1)(3q+1)\)
58. \(4p^2−100\)
59. \(16x^2−72x+81\)
- Answer
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\((4x−9)^2\)
60. \(36y^2+12y+1\)
61. \(8p^2+2\)
- Answer
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\(2(4p^2+1)\)
62. \(81x^2+169\)
63. \(125−8y^3\)
- Answer
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\((5−2y)(25+10y+4y^2)\)
64. \(27u^3+1000\)
65. \(45n^2+60n+20\)
- Answer
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\(5(3n+2)^2\)
66. \(48q^3−24q^2+3q\)
67. \(x^2−10x+25−y^2\)
- Answer
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\((x+y−5)(x−y−5)\)
68. \(x^2+12x+36−y^2\)
69. \((x+1)^3+8x^3\)
- Answer
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\((3x+1)(3x^2+1)\)
70. \((y−3)^3−64y^3\)
Writing Exercises
71. Why was it important to practice using the binomial squares pattern in the chapter on multiplying polynomials?
- Answer
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Answers will vary.
72. How do you recognize the binomial squares pattern?
73. Explain why \(n^2+25\neq (n+5)^2\). Use algebra, words, or pictures.
- Answer
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Answers will vary.
74. Maribel factored \(y^2−30y+81\) as \((y−9)^2\). Was she right or wrong? How do you know?
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?