$s$-th derivative of $p$ at 1 equals $$\sum_{k=0}^{r}(-1)^{r-k} {r \choose k} g_s(k),$$ where $g_s$ is a polynomial of degree $2s$. This equals 0 if $2s<r$.
Fedor Petrov
- 108.9k
- 9
- 264
- 459