Sum over k of r Choose k by -1^r-k by Polynomial

Theorem
Let $r \in \Z_{\ge 0}$.

Then:
 * $\displaystyle \sum_k \binom r k \left({-1}\right)^{r - k} P_r \left({k}\right) = r! \, b_r$

where:
 * $P_r \left({k}\right) = b_0 + b_1 k + \cdots + b_r k^r$ is a polynomial in $k$ of degree $r$.