$\mathcal{I}(\mathcal{Z}(f)) = \langle f \rangle$

Question

Let $f(x) \in k[x]$. Show that $\mathcal{I}(\mathcal{Z}(f)) = \langle f \rangle$ if and only if $f$ is the product of distinct linear factors in $k[x]$.

Here, $\mathcal{Z}$ is the zero locus and $\mathcal{I}$ is the ideal generated. This is a question from Dummit and Foote Chapter 15.1.19.

I am not quite sure where to begin. Any hints/suggestions?