If we have a polynomial and we want to find a factor of it, I know that we should have a root so that P(x)=(x-a)*g(x) , where "a" is the root and P(x) is the given polynomial and g(x) is an other polynomial. And also I know that one of the divisors of the constant term of a polynomial is a root of it.

The question is that the second method for factoring I mentioned is not always valid. Why ? And when it is valid ?

And also is there a proof for it ?

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