$R$ is a finite commutative ring with identity. Let $p(x) \in R[x]$, the ring of polynomials over $R$.
Show that $a \in R$ is a root of $p(x)$ if and only if $p(x)$ can be written as $p(x) = (x-a)g(x)$ with $g(x) \in R[x]$ of degree one less than the degree of $g(x)$.
The "if" part is trivial. I'm struggling with the "only if" part since $R[x]$ is not an Euclidean domain so we cannot use the Division Algorithm. $R$ may not even be an Integral domain, so I can't view $p(x)$ as a polynomial in its field of fractions.
Is there some other way to do it using finiteness of $R$?
Induction on $\deg p$: If $p(x)=a_nx^n+\ldots +a_0\in R[x]$ is a polynomial of degree $n>0$ with root $a$, then so is $q(x):=p(x)-a_nx^{n-1}(x-a)$.