I'm studing theory about polynomial ring, I have this exercise:
For what prime number does the polynomial $f=x^3+\overline{2}x +\overline{2}\in\mathbb{Z}_p$ admit $\overline{3}$ as a root (I hope the term is correct)?
I work in this way $f(\overline{3}) = \overline{27}+\overline{6}+\overline{2}=\overline{35}$ But this is correct if $\overline{35}=\overline{0}$; this happens in $\mathbb{Z}_p$ if and only if: $35\equiv0\pmod p$ So I can say $p=5$. The exercise continues, but I don't know how to proceed:
Write $f$ as product of irreducible factors in $\mathbb{Z}_p$