SOLUTION: How do I factor completely x^8-1?

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Question 152956: How do I factor completely x^8-1?
Found 2 solutions by scott8148, jim_thompson5910:
Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
this factors multiple times using the difference of squares

x^8-1=(x^4-1)(x^4+1)

x^4-1=(x^2-1)(x^2+1)

x^2-1=(x-1)(x+1)

x^8-1=(x^4+1)(x^2+1)(x+1)(x-1)

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E8-1 Start with the given expression


%28x%5E4%29%5E2-%281%29%5E2 Rewrite x%5E8 as %28x%5E4%29%5E2. Rewrite 1 as %281%29%5E2


%28x%5E4%2B1%29%28x%5E4-1%29 Factor using the difference of squares formula. Remember, the difference of squares formula is A%5E2-B%5E2=%28A%2BB%29%28A-B%29


%28x%5E4%2B1%29%28%28x%5E2%29%5E2-%281%29%5E2%29 Rewrite x%5E4 as %28x%5E2%29%5E2. Rewrite 1 as %281%29%5E2.


%28x%5E4%2B1%29%28x%5E2%2B1%29%28x%5E2-1%29 Factor %28x%5E2%29%5E2-%281%29%5E2 using the difference of squares formula


%28x%5E4%2B1%29%28x%5E2%2B1%29%28x%5E2-%281%29%5E2%29 Rewrite 1 as %281%29%5E2.


%28x%5E4%2B1%29%28x%5E2%2B1%29%28x%2B1%29%28x-1%29 Factor x%5E2-%281%29%5E2 using the difference of squares formula


So the expression x%5E8-1 factors to %28x%5E4%2B1%29%28x%5E2%2B1%29%28x%2B1%29%28x-1%29


In other words, x%5E8-1=%28x%5E4%2B1%29%28x%5E2%2B1%29%28x%2B1%29%28x-1%29