Question about how to use Eisenstein's criterion

Question

So I have this exercise:

I have to show if $f:=2X^5-6X+6 \in \mathbb{Z}[X]$ is irreducible in $\mathbb{Z}[X]$.

So Clearly is reducible because $f$ can be written as $f=2(X^5-3X+3)$. But here comes my question:

If I use Eisenstein's criterion with $p=3$ I get that is an irreducible polynomial since $p$ divides each $a_i$ for $0 ≤ i < n$, $p$ does not divide $a_n$, and $p^2$ does not divide $a_0$. Where is my error?

Answer 1

Eisenstein’s criterion only works for primitive polynomials, that is, for polynomials whose coefficients have GCD equal to 1. This is precisely to exclude cases like your example.