# Help to generalize a theorem about roots dividing polynomials

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?

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I'd recommend not accepting an answer so quickly, esp. if parts remain unanswered. Some readers, esp. experts short on time, may only browse questions with unaccepted answers, so we may lose access to their insights by accepting answers very quickly. –  Bill Dubuque Nov 12 '12 at 19:38