I have seen the following theorem for one variable polynomials:
Theorem Let $P \in \mathbb Q[x]$ and $P(\alpha) = 0$ then $(x-\alpha)|P(x)$.
How could it be generalized to multiple variables and other rings?
I have seen the following theorem for one variable polynomials:
Theorem Let $P \in \mathbb Q[x]$ and $P(\alpha) = 0$ then $(x-\alpha)|P(x)$.
How could it be generalized to multiple variables and other rings?
The theorem is true over any coefficient ring since the division algorithm works universally for monic polynomials.
There is no analogous multivariate generalization. However there are multivariate generalizations of the division algorithm, e.g. the Grobner basis algorithm.