Have the following I'm stuck on:
Suppose $p(x)=p_0+p_1 x+p_2 x^2+\cdots+p_n x^n$ is a polynomial of degree $n \geq 1$. Show that if $(x-a)^k$ divides $p(x)$ for some $a\in\mathbb R$ and some integer $k \geq 1$, then $(x-a)^{k-1}$ divides the derivative $p'(x)$. [Any standard results must be clearly stated.] Is the converse of this result true? Justify.
Many thanks in advance.