Theorem: For polynomials modulo $m$, for a number $m≥2$, we have: if $r$ is a root of the polynomial $f$, then division of $f$ by $x-r$ yields remainder $0$. This means: It is possible to write $$f(x) = (x-r)q(x)$$ for some (quotient) polynomial $q$.
This result is for a single variable $x$. I wante to see (if any) a version of this threorem for the case of a multivariate polynomial $f$ with
1) two variables $x,y$.
2) three variables $x,y,z$.