Let $f=x^3+2x \in \mathbb{Z}_3[x]$, $A=\mathbb{Z}_3[x]/(f)$ and $\alpha = [x^2+x+2]_f \in A$.
More specifically I should find $z \in \mathbb{Z}_3[x]$, where the degree of $z$ is $\leq 2$, such that $\alpha [z]_f=[x^2]_f$, or prove that such polynomial doesn't exist.
I found that for $[z]_f=[x]_f$, $[x^2 + x+2]\cdot[x]=[x^3+x^2+2x]=[x^2]_f$.
However I also thought that $\alpha\cdot[z]_f=[x^2]_f$ $\Leftrightarrow$ $\alpha|[x^2]_f$.
Is that true?