Proof of Polynomial Statement using Taylor Polynomials

Question

If we have two polynomials $f,g$ where $\deg(f),\deg(g)\le n$ and all of their derivatives up to $n$ of $x_0$ are equal, then $f=g$. I think that we have to use Taylor Polynomials to solve this. I already used that the $k$^th derivative of the Taylor Polynomial of $x_0$ of $f$ is equal to the equivalent of $g$. However, I am not sure how to proceed in order to come to the conclusion that $f=g$.

Answer 1

Use the fact that, since both $f$ and $g$ are polynomial functions whose degree is at most $n$, then$$f(x)=\sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k\text{ and }g(x)=\sum_{k=0}^n\frac{g^{(k)}(x_0)}{k!}(x-x_0)^k.$$