Let F be a Field. $f \in F[X] \text{ and } h\in F$. Show that $(x-h)^n|f\text{ if and only if }f(h)=f'(h)=...=f^{n-1}(h)=0$
I want to prove it using $f(h)=0 \Leftrightarrow (x-h)|f$ which i already managed to prove. My Idea was to prove it via induction. The base case for n=1 is easily shown using what i stated above. But i have problems with the induction step.