Showing that a Polynomial is Irreducible Over $\mathbb{Z}_3$

Question

What is the most "efficient" way of showing that a polynomial is irreducible over $\mathbb{Z}_3$? In particular, how do I show $p(x) = x^3 + x^2 + 2$ is irreducible over the field $\mathbb{Z}_3$? The only way I can think of is the brute force method, and there has to be a smarter solution, right? Thank you.

Answer 1

Check it has no root. As in $\mathbf Z/3\mathbf Z$, we have $x^3=x$ (this is lil' Fermat), it means $x^2+x-1$ has no root, you just have to set $x=1, -1,0$ successively.