Polynomials over field are Euclidean domain. So for any $a$
$f(x)=(x-a)Q(x) + R(a)$ where $R(a) = f(a)$.
So we have a corollary: $a$ is a root of polynomial $f(x) \iff (x-a)|f(x)$.
Is this true for polynomials over rings? For example let $\mathbb F[x_1, x_2, ..., x_n]$ be a polynomial of several variables. We can also see it as a ring of polynomials over ring: $\mathbb F[x_2, ..., x_n][x_1]$. If we keep $deg(f)$ as a Euclidean norm is this true that remainder norm is less than norm of divider?