maclaurin's theorem

States that for any polynomial $P(x)$ and any function $f(x)$:

If $P^k(x_0)=f^k(x_0)$ for all $k=1,2,...n$ then $P(x)$ is the nth degree Taylor Polynomial centered at $x_o$ of $f(x)$

In simple terms: each term of a taylor polynomial must be equal to the nth derivative of the original function when you sub in some constant $x_0$