Factoring (x^n-a^n) with (x^2-a^2): Is it Possible?

[applegatecz]

Homework Statement

I would like to factor (x^n-a^n) such that (x^2-a^2) is one of the factors. Is this possible?

Homework Equations

The Attempt at a Solution

I tried to get this with a kind of reverse polynomial long division, but couldn't do it.

 

[D H]

This factoring is possible for some cases of n. It is not true in general. (x^2-a^2) is equal to (x-a)(x+a). It certainly is the case that (x-a) divides (x^n-a^n). What is the quotient? Under what circumstances does (x+a) divide this quotient?