I'll assume . For it can be checked directly.

Let

Then

Differentiating more, it's easily seen that

Where on the left side we have differentiations and on the right side are polynomials of degree .

As such, form a basis to the vector space of polynomials of degree at most . Therefore,

is a linear combination of

up to times differentiation.

But since the zero of at is of order , we have that all the functions in (1) evaluated at vanish, and thus their linear combination evaluated at that point,

also does.