Writing $f(x)=h(x)\cdot(x-\alpha)^m$ when $\alpha$ is a root of multiplicity $m$

Question

Let $f$ be a real function, and $\alpha \in \Bbb R$ a root of $f$ of multiplicity $m$. Under what conditions can I write $f(x)=h(x)\cdot(x-\alpha)^m$, where $h(\alpha)\neq 0$ ?

I know it's true if $f$ is a polynomial, or when $f$ is an analytic function. But is it true in other cases as well?