Question

How do you factor `x^4 + 64`?

Answer 1

`x^4+64 = (x^2-4x+8)(x^2+4x+8)`

Explanation:

Given:

`x^4+64`

Note that this is positive and therefore non-zero for any real value of `x`. So it has no linear factors with real coefficients. We can find quadratic factors.

We can use the difference of squares identity to help with this:

`A^2-B^2=(A-B)(A+B)`

First note that:

`(x^2+8)^2 = x^4+16x^2+64 = x^4+64+(4x)^2`

So:

`x^4+64 = (x^2+8)^2-(4x)^2`

`color(white)(x^4+64) = ((x^2+8)-4x)((x^2+8)+4x)`

`color(white)(x^4+64) = (x^2-4x+8)(x^2+4x+8)`