Let $R$ be a conmutative ring with identity, let $\displaystyle f=\sum_{k=0}^{n}a_k x^k \in R[x]$ and $r\in R$. If $f=(x-r)^m g$, $m\in\mathbb{N}$ and $g\in R[x]$ with $g(r)\neq 0$, then the root $r$ is said to have $multiplicity\,\,\, m$. If the multiplicity is 1 the root is called simple.
How to prove that the above definition of the multiplicity of a root is well defined?