Let $P(x)$ be a polynomial in $\mathbb{Z}[x]$ of degree 5 such that $P(1)=3$ and $P=(x-1)^5\bmod 3$. Show that as a polynomial in $\mathbb{Q}[x]$, $P(x)$ is irreducible.
So here I find that $$P(x)=(x-1)^5+3=x^5-5x^4+10x^3-10x^2+5x+2.$$ How can I show this irreducible? Can I use Eisenstein's Criterion? How would I do that? For instance, if my prime is 5 then it divides 4 of the 6 terms but what about those other 2?