How to factor $(x^5+1) (x^5-1) $

Question

I have this:

$ (x^5+1) (x^5-1) $

And I don't know how to continue factor.

Geogebra's Factor says:

$(x+1)(x-1)(x^4-x^3+x^2-x+1)(x^4+x^3+x^2+x+1)$

Answer 1

Since $-1$ is a root of $x^5+1$, we know that it is divisible by $x+1$. Likewise, since $1$ is a root of $x^5-1$, we know that it is divisible by $x-1$. You can use polynomial long division to obtain the other factors.

Answer 2

Hint:

$$x^n-1=(x - 1)[x^{n - 1} + x^{n - 2} + ... + x^2 + x + 1] $$

Answer 3

I assume that you are trying to factor these polynomials over $\mathbb{Q}$.

What you need is to check that $f(x)=x^{p-1}+\ldots+1$ is irreducible for any prime $p$. To see this, consider $f(x+1)=\frac{(x+1)^p-1}{x}$ for which Eisenstein's criterion works.