How to prove irreducibility of $P(x)=x^4-2x^3-2x^2+2x-1$ over $\mathbb{Q}$?

Question

I have polynomial $P(x)=x^4-2x^3-2x^2+2x-1$ and I have strong suspicions it's irreducible. Any ideas how to prove that?

Thank you.

Answer 1

The polynomial is irreducible over $\Bbb Z$ and $\Bbb Q$, because it is already irreducible modulo $3$, so over $\Bbb F_3$, which is easy to see.

Alternatively, you can use Eisenstein with a shift $x\mapsto x+1$, or just see it directly, like in this post: Show that polynomial is irreducible over $\mathbb{Q}$