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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

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up vote 2 down vote accepted

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.

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The reason I ask about multiple variables is situations such as this where I said that a polynomial in x with coefficients in Q(m,n) is divisible by x-1: math.stackexchange.com/questions/228907/solutions-of-a2b2-c2/… - I hoped for a better theoretical basis of this. –  sperners lemma Nov 12 '12 at 17:12
    
@spernerslemma I recommend that you edit your questions to say more about what type multivariate of generalizations that you seek. –  Bill Dubuque Nov 12 '12 at 18:41

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