Section 2.7
Miscellaneous Factoring Problems
Factor each of the following completely, or identify the expression as being prime.\begin{array}{l l l}
1. \;\;{{x}^{2}}+14x-120\;\;& 2. \;\;5{{x}^{2}}+13x+6\;\;& 3. \;\;1-64{{x}^{2}}\;\;\\
4. \;\;3{{x}^{3}}-66{{x}^{2}}+363x\;\; & 5. \;\;24{{z}^{2}}+13z-45\;\;& 6. \;\;8{{a}^{2}}-2a-1\;\;\\
7. \;\;2{{x}^{4}}-17{{x}^{2}}-9\;\;& 8. \;\;{{x}^{2}}+36\;\; & 9. \;\;5{{a}^{2}}+12ab+4{{b}^{2}}\;\;\\
10. \;\;1-a-b+ab\;\;& 11. \;\;rx+rz\;\; & 12. \;\;{{(m+n)}^{2}}-4(m+n)+4\;\;\\
13. \;\;{{x}^{2}}+7x-12\;\;& 14. \;\;49{{x}^{2}}-168x+144\;\;& 15. \;\;{{x}^{4}}-16\;\;\\
16. \;\;5x+15-ax-3a\;\;& 17. \;\;3{{x}^{2}}-27\;\;& 18. \;\;{{x}^{2}}-13x-48\;\;\\
19. \;\;2{{x}^{2}}+8x+8\;\;& 20. \;\;2{{x}^{2}}-13x-7\;\;& 21. \;\;6{{x}^{2}}-19x+15\;\;\\
22. \;\;14{{x}^{2}}-49x-28\;\;& 23. \;\;5x(x+3)-7(x+3)\;\;& 24. \;\;4{{x}^{3}}y-x\;\;\\
25. \;\;3{{x}^{2}}+12\;\;& 26. \;\;5{{x}^{2}}y-5xy-10y\;\;& 27. \;\;7a(4a+b)-6(4a+b)\;\;\\
28. \;\;{{x}^{2}}(x+1)+x(x+1)\;\;& 29. \;\;{{x}^{2}}+10x-24\;\;& 30. \;\;4x\left( {{y}^{2}}-1 \right)+12{{x}^{2}}\left( {{y}^{2}}-1 \right)\;\;\\
31. \;\;2{{x}^{2}}-18{{y}^{4}}\;\;& 32. \;\;ax+2a+bx+2b\;\;& 33. \;\;{{(x+y)}^{2}}-3x-3y\;\;\\
34. \;\;{{x}^{2}}-4xy+4{{y}^{2}}-9\;\;& 35. \;\;20p-9{{p}^{2}}+{{p}^{3}}\;\;& 36. \;\;\left( {{x}^{2}}-{{y}^{2}} \right)+{{\left( x-y \right)}^{2}}\;\;\\
37. \;\;3ax+2-6a-x\;\;& 38. \;\;3{{x}^{2}}+4x+5\;\;& 39. \;\;9{{x}^{2}}+{{y}^{2}}-6xy-1\;\;\\
40. \;\;{{x}^{2}}-x-72\;\;& 41. \;\;169{{a}^{4}}-81{{(a+1)}^{2}}\;\;& 42. \;\;4{{a}^{4}}{{b}^{2}}-12{{a}^{2}}bc-72{{c}^{2}}\;\;\\
43. \;\;9+{{(x+K)}^{2}}\;\;& 44. \;\;1-{{(x+3K)}^{3}}\;\;& 45. \;\;{{(x+3)}^{2}}-6y(x+3)-7{{y}^{2}}\;\;\\
46. \;\;{{x}^{2}}-2xy-2x+4y\;\;& 47. \;\;{{x}^{2}}-5x-14\;\;& 48. \;\;{{(2x-1)}^{2}}-2(x-2)(2x-1)\;\;\\
49. \;\;2{{x}^{2}}y-4x{{y}^{2}}\;\;& 50. \;\;2{{x}^{2}}+3x+1\;\;& 51. \;\;(x-1)(x+2)+5(x+2)\;\;\\
52. \;\;27{{x}^{3}}+8{{y}^{3}}\;\;& 53. \;\;3{{x}^{2}}-7x+2\;\;& 54. \;\;6{{x}^{3}}-3{{x}^{2}}-24x+12\;\;\\
55. \;\;36-{{(x-K)}^{2}}\;\;& 56. \;\;15{{x}^{2}}+x-6\;\;& 57. \;\;4{{x}^{2}}-{{y}^{2}}+10y-25\;\;\\
58. \;\;(x+2)(x+4)(x-7)\;\;& 59. \;\;8{{x}^{2}}+6x-9\;\;& 60. \;\;3ax-3b{{x}^{2}}+7a-7bx\;\;\\
61. \;\;3{{x}^{2}}-6xy\;\;& 62. \;\;x^2-6x+9\;\;& 63. \;\;4x^2-4x+1-9y^2\;\;\\
64. \;\;7{{x}^{2}}-14x-21\;\;& 65. \;\;5{{x}^{2}}-125\;\;& 66. \;\;{{x}^{2}}-11x+30\;\;\\
67. \;\;12{{x}^{3}}-4{{x}^{2}}\;\;& 68. \;\;c(x+m)=x(a+b)\;\;& 69. \;\;(x+T)^2-4(x+T)-5\;\;\\
70. \;\;7{{x}^{4}}-567\;\;& 71. \;\;2m{{a}^{2}}+{{a}^{2}}-2m{{b}^{2}}-{{b}^{2}}\;\;& 72. \;\;12{{x}^{2}}-8x-15\;\;\\
73. \;\;{{(a+b)}^{2}}+8(a+b)+16\;\;& 74. \;\;9{{p}^{2}}-13p+4\;\;& 75. \;\;40{{x}^{2}}+43x-6\;\;\\
76. \;\;3{{p}^{3}}-{{p}^{2}}-10p\;\;& 77. \;\;12{{x}^{2}}-28x+15\;\;& 78. \;\;{{(a+2t)}^{2}}-{{(2a-5t)}^{2}}\;\;\\
79. \;\;10{{x}^{2}}-89x-9\;\;& 80. \;\;100{{x}^{2}}-9\;\;& 81. \;\;{{a}^{2}}-6ab+9{{b}^{2}}-36{{x}^{2}}\;\;\\
82. \;\;{{a}^{4}}-\left[ 4{{a}^{2}}-12a+9 \right]\;\;& 83. \;\;4{{a}^{4}}-37{{a}^{2}}+9\;\;& 84. \;\;(a-1)\left( {{a}^{2}}-2 \right)+a\cdot (a-1)\;\;\\
85. \;\;{{a}^{3}}-{{b}^{6}}\;\;& 86. \;\;6{{x}^{2}}-36x+54\;\;& 87. \;\;{{x}^{2}}+ax-ay-{{y}^{2}}\;\;\\
88. \;\;{{x}^{3}}+{{y}^{3}}\;\;& 89. \;\;10{{x}^{2}}-21x-10\;\;& 90. \;\;{{(6t-3K)}^{2}}-{{(2t+K)}^{2}}\;\;\\
91. \;\;{{\left( {{x}^{2}}+2x \right)}^{3}}+1\;\;& 92. \;\;18{{x}^{2}}+9x-5\;\;& 93. \;\;{{x}^{2}}+3xy-kx-3ky\;\;\\
94. \;\;{{x}^{4}}+1\;\;& 95. \;\;48{{x}^{2}}-22x-15\;\;& 96. \;\;{{x}^{2}}(4y+3)-4y-3\;\;\\
97. \;\;{{\left( {{x}^{2}}-1 \right)}^{2}}-{{(x-1)}^{2}}\;\;& 98. \;\;{{x}^{4}}-3{{x}^{3}}-4{{x}^{2}}\;\;& 99. \;\;2{{a}^{3}}-16\;\;\\
100. \;\;{{(5a+b)}^{2}}-{{(2a-b)}^{2}}\;\;& 101. \;\;{{x}^{4}}-2{{x}^{2}}+1\;\;& 102. \;\;{{x}^{4}}-4{{x}^{2}}+4x-1\;\;
\end{array}
103. Consider the expression ${{x}^{2}}+Kx-36$. There are nine different values of $K$ (note that $K$ does not have to be positive) for which this expression factors. Determine those nine values. Then write the expression (with the appropriate value of $K$ used) in each case, as well as the factorization of the expression.
104. Consider the expression ${{x}^{2}}+Kx-48$. There are ten different values of $K$ (note that $K$ does not have to be positive) for which this expression factors. Determine those ten values. Then write the expression (with the appropriate value of $K$ used) in each case, as well as the factorization of the expression.
105. Factor completely:
a. $\left( {{x}^{2}}-5x+6 \right)(x+4)+\left( {{x}^{2}}+3x-10 \right)(x-3)$
b. $\left( {{x}^{2}}+x-6 \right)(2x+1)-\left( {{x}^{2}}-3x+2 \right)(x-2)$
106. Consider the formula $P={{n}^{2}}-n+41$.
a. Compute the value of P that is obtained when $n = 1,\; 2,\; 3,\; 4,\; 5,\; 6,\; 7,\; 8,\; 9,\; 10.$
b. Show that when $n=41$, then $P$ is not a prime number.
c. Show, as simply as possible, that when $n=42$ and again when $n=45$, then $P$ is not a prime number.
d. Determine another value of $n$ larger than $n=45$ for which $P$ is not prime.
107. Consider the formula $P={{n}^{3}}-n+41$.
a. Show that when $n=1$ and $n=2$, this expression takes on a value that is prime.
b. Find three values of $n$ for which this does not take on a prime value.