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

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

Then:
 * $\ds \sum_k \binom r k \paren {-1}^{r - k} \map {P_r} k = r! \, b_r$

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