Prove that a polynomial P ∈ R [ X ] {\displaystyle {}P\in \mathbb {R} [X]} has degree ≤ d {\displaystyle {}\leq d} (or it is P = 0 {\displaystyle {}P=0} ), if and only if the ( d + 1 ) {\displaystyle {}(d+1)} -th derivative of P {\displaystyle {}P} is the zero poynomial.
![{\displaystyle {}P\in \mathbb {R} [X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bf6bcf64e16c3ac6b30ecd6577e72648ab2325a)
