Polynomial function/R/(X-a)^n divides/Zero of derivative/Exercise

Let ${}P\in \mathbb {R} [X]$ be a polynomial, ${}a\in \mathbb {R}$ and ${}n\in \mathbb {N}$.

Prove that ${}P$ is a multiple of ${}(X-a)^{n}$ if and only if ${}a$ is a zero of all the derivatives ${}P,P^{\prime },P^{\prime \prime },\ldots ,P^{(n-1)}$.

Create a solution