Suppose we have a polynomial over the integers with a non-zero leading coefficient over $\mod p$. Suppose $r$ is a zero of $f(r)$ is congruent to $0 \mod p$, show there exists polynomial $g(x)$ such that $(x-r)(g(x)) = a_nx^n..+a_1x +b_o$ where $a_0$ is congruent to $b_0 \mod p$.
This is what I have so far, since $f(r)=0$, then I know I can write $f(x)=(x-r)(g(x))$.
I know (by doing an example, that the product of $r$ and $b_0$ is gonna be congruent to $a_0 \mod p$. But that's all I have so far. Can anyone help?