Factor Theorem in Unique Factorization Domain

Question

To Prove: Suppose $r \in R$ and $P(x)$ a polynomial in $R[x]$ where R is a unique factorization domain. Prove that $x-r$ divides $P(x)$ iff $P(r)=0.$ I am able to prove this if R is a field because I can use the Division Algorithm but I am really stuck on this. Help would be greatly appreciated

Answer 1

The Division Algorithm works with no change for polynomials over an integral domain, if we are dividing by a monic polynomial.

Added:

Theorem: Let $D$ be an integral domain with unity, and let $a(x)$, $b(x)$ be polynomials with coefficients in $D$, where $q(x)$ is monic. Then there exist polynomials $q(x)$, $r(x)$ with coefficients in $D$ such that $a(x)=q(x)b(x)+r(x)$, where $r(x)$ is the zero polynomial or has degree less than the degree of $b(x)$.

The proof is basically the same as the proof for $D$ a field. For example we can consider the collection $P$ of all polynomials over $D$ of the form $a(x)-f(x)b(x)$, and let $r(x)$ have smallest degree among these. It is easy to show that if the degree of $r(x)$ is greater than or equal to the degree of $b(x)$, there is a polynomial in $P$ of degree smaller than the degree of $r(x)$. It is at this stage where the fact that $b(x)$ is monic is needed.