Question

How do you factor `x^8-256`?

Answer 1

Remember the special products:
`(A-B)(A+B)=A^2-B^2` and vice versa.

`x^8-256=(x^4)^2-(16)^2=(x^4-16)(x^4+16)`

You can't factor the second term any further, but the first term can be factored according to the same rule:
`x^4-16=(x^2)^2-(4)^2=(x^2-4)(x^2+4)`

And again:
`x^2-4=(x-2)(x+2)`

If we put all of this together we get:

`x^8-256=(x-2)(x+2)(x^2+4)(x^4+16)`