# Polynomfunktion/R/(X-a)^n teilt/Ableitungsnullstelle/Aufgabe/en

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Let ${\displaystyle {}P\in \mathbb {R} [X]}$ be a polynomial ${\displaystyle {}a\in \mathbb {R} }$ and ${\displaystyle {}n\in \mathbb {N} }$. Prove that ${\displaystyle {}P}$ is a multiple of ${\displaystyle {}(X-a)^{n}}$ if and only if ${\displaystyle {}a}$ is a zero of all the derivatives ${\displaystyle {}P,P^{\prime },P^{\prime \prime },\ldots ,P^{(n-1)}}$.