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Question 732874: What are the factors, (5 at minimum,) of 16x^9-16x?
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!
16x9 - 16x
Factor out 16x
16x(x8 - 16)
The parentheses contains the difference of 2 perfect squares:
16x(x4 - 4)(x4 + 4)
The parentheses contains the difference of 2 perfect squares:
16x(x2 - 2)(x2 + 2)(x4 + 4)
The 3rd parentheses can become the difference of perfect squares
if we play the trick of adding and subtracting 4x2
16x(x2 - 2)(x2 + 2)(x4 + 4 + 4x2 - 4x2)
Swap the 2nd and 3rd terms in the 3rd parentheses
16x(x2 - 2)(x2 + 2)(x4 + 4x2 + 4 - 4x2)
Change the third parentheses to brackets and enclose its
first three terms in parentheses
16x(x2 - 2)(x2 + 2)[(x4 + 4x2 + 4) - 4x2]
Factor the trinomial as a perfect square
16x(x2 - 2)(x2 + 2)[(x2 + 2)2 - 4x2]
Now the bracket contains the difference of 2 perfect squares:
16x(x2 - 2)(x2 + 2)[(x2 + 2) - 2x][(x2 + 2) + 2x]
Remove the parentheses inside the brackets:
16x(x2 - 2)(x2 + 2)[x2 + 2 - 2x][x2 + 2 + 2x]
Arrange the trinmials in brackets in descending order and
change the brackets to parentheses:
16x(x2 - 2)(x2 + 2)(x2 - 2x + 2)(x2 + 2x + 2)
Here are 105 factors of that, pick any 5.
1. 1
2. 2
3. 4
4. 8
5. 16
6. x
7. (x2-2)
8. (x2+2)
9. (x2-2x+2)
10. (x2+2x+2)
11. x(x2-2)
12. x(x2+2)
13. (x2-2)(x2+2)
14. 2x(x2-2)
15. x(x2+2)
16. (x2-2)(x2+2)
17. 4x(x2-2)
18. x(x2+2)
19. (x2-2)(x2+2)
20. 8x(x2-2)
21. x(x2+2)
22. (x2-2)(x2+2)
23. 16x(x2-2)
24. x(x2+2)
25. (x2-2)(x2+2)
26. x(x2-2)(x2+2)
27. x(x2-2)(x2-2x+2)
28. x(x2-2)(x2+2x+2)
29. x(x2+2)(x2-2x+2)
30. x(x2+2)(x2+2x+2)
31. x(x2-2x+2)(x2+2x+2)
32. (x2-2)(x2+2)(x2-2x+2)
33. (x2-2)(x2+2)(x2+2x+2)
34. (x2-2)(x2-2x+2)(x2+2x+2)
35. (x2+2)(x2-2x+2)(x2+2x+2)
36. 2x(x2-2)(x2+2)
37. x(x2-2)(x2-2x+2)
38. x(x2-2)(x2+2x+2)
39. x(x2+2)(x2-2x+2)
40. x(x2+2)(x2+2x+2)
41. x(x2-2x+2)(x2+2x+2)
42. (x2-2)(x2+2)(x2-2x+2)
43. (x2-2)(x2+2)(x2+2x+2)
44. (x2-2)(x2-2x+2)(x2+2x+2)
45. (x2+2)(x2-2x+2)(x2+2x+2)
46. 4x(x2-2)(x2+2)
47. x(x2-2)(x2-2x+2)
48. x(x2-2)(x2+2x+2)
49. x(x2+2)(x2-2x+2)
50. x(x2+2)(x2+2x+2)
51. x(x2-2x+2)(x2+2x+2)
52. (x2-2)(x2+2)(x2-2x+2)
53. (x2-2)(x2+2)(x2+2x+2)
54. (x2-2)(x2-2x+2)(x2+2x+2)
55. (x2+2)(x2-2x+2)(x2+2x+2)
56. 8x(x2-2)(x2+2)
57. x(x2-2)(x2-2x+2)
58. x(x2-2)(x2+2x+2)
59. x(x2+2)(x2-2x+2)
60. x(x2+2)(x2+2x+2)
61. x(x2-2x+2)(x2+2x+2)
62. (x2-2)(x2+2)(x2-2x+2)
63. (x2-2)(x2+2)(x2+2x+2)
64. (x2-2)(x2-2x+2)(x2+2x+2)
65. (x2+2)(x2-2x+2)(x2+2x+2)
66. 16x(x2-2)(x2+2)
67. x(x2-2)(x2-2x+2)
68. x(x2-2)(x2+2x+2)
69. x(x2+2)(x2-2x+2)
70. x(x2+2)(x2+2x+2)
71. x(x2-2x+2)(x2+2x+2)
72. (x2-2)(x2+2)(x2-2x+2)
73. (x2-2)(x2+2)(x2+2x+2)
74. (x2-2)(x2-2x+2)(x2+2x+2)
75. (x2+2)(x2-2x+2)(x2+2x+2)
76. x(x2-2)(x2+2)(x2-2x+2)
77. x(x2-2)(x2+2)(x2+2x+2)
78. x(x2-2)(x2-2x+2)(x2+2x+2)
79. x(x2+2)(x2-2x+2)(x2+2x+2)
80. (x2-2)(x2+2)(x2-2x+2)(x2+2x+2)
81. 2x(x2-2)(x2+2)(x2-2x+2)
82. x(x2-2)(x2+2)(x2+2x+2)
83. x(x2-2)(x2-2x+2)(x2+2x+2)
84. x(x2+2)(x2-2x+2)(x2+2x+2)
85. (x2-2)(x2+2)(x2-2x+2)(x2+2x+2)
86. 4x(x2-2)(x2+2)(x2-2x+2)
87. x(x2-2)(x2+2)(x2+2x+2)
88. x(x2-2)(x2-2x+2)(x2+2x+2)
89. x(x2+2)(x2-2x+2)(x2+2x+2)
90. (x2-2)(x2+2)(x2-2x+2)(x2+2x+2)
91. 8x(x2-2)(x2+2)(x2-2x+2)
92. x(x2-2)(x2+2)(x2+2x+2)
93. x(x2-2)(x2-2x+2)(x2+2x+2)
94. x(x2+2)(x2-2x+2)(x2+2x+2)
95. (x2-2)(x2+2)(x2-2x+2)(x2+2x+2)
96. 16x(x2-2)(x2+2)(x2-2x+2)
97. x(x2-2)(x2+2)(x2+2x+2)
98. x(x2-2)(x2-2x+2)(x2+2x+2)
99. x(x2+2)(x2-2x+2)(x2+2x+2)
100. (x2-2)(x2+2)(x2-2x+2)(x2+2x+2)
101. x(x2-2)(x2+2)(x2-2x+2)
102. 2x(x2-2)(x2+2)(x2-2x+2)
103. 4x(x2-2)(x2+2)(x2-2x+2)
104. 8x(x2-2)(x2+2)(x2-2x+2)
105. 16x(x2-2)(x2+2)(x2-2x+2)
Edwin
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