Factorization of a polynomial into irreducible factors

Question

$f(x)$ = $x^5$+$2x^2$+$2x$+$2$ $ \in Z_3[x]$

How do i factorize it into irreducible factors ?

Any hints are welcome.

Answer 1

Hint: If $i=\sqrt{-1}$ in an extension field of $\Bbb{Z}_3$, then $$ f(i)=i^5+2(i^2+1)+2i=i+0+2i=3i=0. $$