Let P ∈ R [ X ] {\displaystyle {}P\in \mathbb {R} [X]} be a polynomial, a ∈ R {\displaystyle {}a\in \mathbb {R} } and n ∈ N {\displaystyle {}n\in \mathbb {N} } . Prove that P {\displaystyle {}P} is a multiple of ( X − a ) n {\displaystyle {}(X-a)^{n}} if and only if a {\displaystyle {}a} is a zero of all the derivatives P , P ′ , P ′ ′ , … , P ( n − 1 ) {\displaystyle {}P,P^{\prime },P^{\prime \prime },\ldots ,P^{(n-1)}} .
![{\displaystyle {}P\in \mathbb {R} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bf6bcf64e16c3ac6b30ecd6577e72648ab2325a)
