Let $R$ be a commutative ring with unity, $a\in R$, $f(x) \in R[x]$. Then $a$ is a zero of $f$ iff $x-a$ is a factor of $f$.
Solution: If $a$ is a zero of $f$, then by division algorithm, we can write $f(x) = (x-a)g(x) + r(x)$ such $\deg (x-a) > \deg r$ or $\deg r = 0$ and $g(x) \in R[x]$. But notice $r(x) = f(x) - g(x)(x-a) \Rightarrow r(a) = 0$ Therefore $f(x) = g(x)(x-a)$.
the other direction is trivial.
Is this approach correct? can someone give me feedback?