SOLUTION: x^12 - 1 Factor.

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Question 8167: x^12 - 1
Factor.
Found 2 solutions by longjonsilver, Earlsdon:
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
this is a difference of 2 squares, if written as %28x%5E6%29%5E2+-+1%5E2...so we can write this as %28x%5E6%2B1%29%28x%5E6-1%29

jon.

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Ok,this gets a little messy, so please bear with me.
Factor: x%5E12+-+1 Start with the difference of two squares.
x%5E12+-+1+=+%28x%5E6+-+1%29%28x%5E6+%2B+1%29 NB: x%5E12+=+%28x%5E6%29%5E2
Take the first factor: %28x%5E6+-+1%29 and note that it is also the difference of two squares.
%28x%5E6+-+1%29+=+%28x%5E3+-+1%29%28x%5E3+%2B+1%29
Notice here that the 1st factor %28x%5E3+-+1%29 is the difference of two cubes and that the second factor %28x%5E3+%2B+1%29 is the sum of two cubes. These can be factored using the formulas:
%28A%5E3+-+B%5E3%29+=+%28A+-+B%29%28A%5E2+%2B+AB+%2B+B%5E2%29 and;
%28A%5E3+%2B+B%5E3%29+=+%28A+%2B+B%29%28A%5E2+-+AB+%2B+B%5E2%29
So, we have:
%28x%5E3+-+1%29+=+%28x+-+1%29%28x%5E2+%2B+x+%2B+1%29 and;
%28x%5E3+%2B+1%29+=+%28x+%2B+1%29%28x%5E2+-+x+%2B+1%29
Now, going back to the first line, let's look at the 2nd factor: %28x%5E6+%2B+1%29 and factor this:
%28x%5E6+%2B+1%29+=+%28x%5E2%2B+1%29%28x%5E4+-+x%5E2+%2B+1%29
Now we can put it all together:
Whew!
You can verify this by multiplying all of these factors to see that you get x%5E12+-+1%29 I'll leave this as an exercise for the student.
A final note:
I have left the factor %28x%5E2+%2B+1%29 in its unfactored form, not because it can't be factored, but because I don't know whether or not you have covered complex numbers yet. But, in case you have, here's how you would factor x%5E2+%2B+1
%28x%5E2+%2B+1%29+=+%28x+-+i%29%28x+%2B+i%29 where: i+=+sqrt%28-1%29
Using FOIL to check:
%28x+-+i%29%28x+%2B+i%29+=+%28x%5E2+%2B+xi+-+xi+-+i%5E2%29 NB: i%5E2+=+%28sqrt%28-1%29%29%5E2 = -1
%28x+-+i%29%28x+%2B+i%29+=+x%5E2+-+%28-1%29 = x%5E2+%2B+1%29