Let $R$ be a commutative ring and $f\in R[x]$ be such that $f(r)=0$ for some $r\in R$. I wonder if $x-r$ divides $f(x)$ in this case.
I know this is true when $R$ is an Euclidean domain, since we can write $f(x)=(x-r)g(x)+c$. But is this true for a more general commutative ring $R$?
Plus, if this is false, is there a condition on $R$ weaker than the Euclidean domain that makes this true (PID, UFD)?
Just realize that $f(x)-f(r)$ is a linear combination of $x^k-r^k$'s and $x-r$ divides every $x^k-r^k$.
Indeed, over any commutative ring, if $m$ is a monic polynomial then for each $f$ there is a unique $q$ with $f-qm$ having degree less than that of $m$. Here for $m(x)=x-r$ you get $f(x)=(x-r)q(x)+c$ and $c$ must be $f(r)$.
Yes, this is true and can be stated slightly more abstractly. Here's a proof that doesn't use the division algorithm.
Lemma. Let $R$ be a unital commutative ring and $R[x]$ be the one-variable polynomial ring over $R$. For $\alpha \in R$, let \begin{align*} \varphi = \text{eval}_\alpha: R[x] &\to R\\ x &\mapsto \alpha \end{align*} be the evaluation homomorphism. Then $\ker(\varphi) = (x-\alpha)$ and the induced map $\overline{\varphi}: R[x]/(x-\alpha) \to R$ is an isomorphism.
Note that $\ker(\varphi) = \{f \in R[x] : f(\alpha) = 0\}$.
Proof. Certainly $(x - \alpha) \subseteq \ker(\varphi)$, so it remains to show the reverse inclusion. Given $f = \sum_{i=0}^n a_i x^i \in \ker(\varphi)$, then $$ 0 = f(\alpha) = \sum_i a_i \alpha^i \, . $$ We can rewrite $f$ as a polynomial in $x-\alpha$ by writing $x = (x - \alpha) + \alpha$ and using the binomial theorem: \begin{align*} f &= \sum_{i=0}^n a_i x^i = \sum_{i=0}^n a_i ((x - \alpha) + \alpha)^i = \sum_{i=0}^n a_i \sum_{j = 0}^i \binom{i}{j} (x-\alpha)^j \alpha^{i-j}\\ &= \sum_{i=0}^n a_i \sum_{j = 0}^n \binom{i}{j} (x-\alpha)^j \alpha^{i-j} = \sum_{j = 0}^n \underbrace{\left(\sum_{i=0}^n a_i \binom{i}{j} \alpha^{i-j}\right)}_{b_j} (x-\alpha)^j = \sum_{j=0}^n b_j (x - \alpha)^j \end{align*}
(Note that $\binom{i}{j} = 0$ for $j > i$, which allows us to change the upper limit of the inner sum from $i$ to $n$.) Since $f(\alpha) = 0$, then $$ b_0 = \sum_{i=0}^n a_i \binom{i}{0} \alpha^{i-0} = \sum_{i=0}^n a_i \alpha^i = f(\alpha) = 0 \, . $$ Since $b_0 = 0$, then every term in $f = \sum_{j=0}^n b_j (x - \alpha)^j$ has a factor of $x - \alpha$, so $f \in (x - \alpha)$.
This idea also works in multivariable polynomial rings; see here for the proof.
