Irreducibility of a polynomial in $ \Bbb Q[x]$

Question

I took my friend's notes and there was a question which was asking if $$x^5+9x^4+12x^2+6$$ is irreducible in $ \Bbb Q[x]$. answer was "yes, because it is irreducible in $Z_3$[x]" But isnt $0\in Z_3$ a root of this polynomial? I think it is irreducible because Eisenstein's. (which is for p=3) am I wrong?

Answer 1

You are right, Eisenstein does it. And the polynomial is not irreducible over $\mathbb{Z}_3[x]$, it factors for example as $x\cdot x^4$.