$P(x)=-x^5+10x^3-20x+12$ is irreducible

Question

I would like to prove that $P(x)=-x^5+10x^3-20x+12$ is irreducible over $\Bbb{Q}$

I tried using Eisenstein's criterion, reduction and factorizing but I didn't succeed, any idea ?

Answer 1

By the rational root theorem, $f(x)$ has no rational roots. For $f(p/q)=0$ would imply $p\mid 12$ and $q\mid 1$, and no such value works. By the Gauss Lemma it is enough to show that $f(x)$ is irreducible over $\mathbb{Z}$. Then $$ f(x)=(x^3+ax^2+bx+c)(-x^2+dx+e) $$ leads to a contradiction over $\mathbb{Z}$.