SOLUTION: What are the factors, (5 at minimum,) of 16x^9-16x?

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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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