Let $f=a_{0}+a_{1}x+ ...+ a_{n}x^n$ be a polynomial in $Q[x]$
$f'=a_{1}+2a_{2}x+...+na_{n}x^{n-1}$
Show if $f$ is divisible by $(x-c)^2$ for some $c \in Q$ then $f'(c)=0$
My own thoughts: If $(x-c)^2$ divides $f$ then the remainder is 0 after division. But don't know how to start this really..
If $f$ is divisible by $(x-c)^2$, then $f(x)=(x-c)^2g(x)$ for some polynomial $g(x)$, as suggested in the comments. Taking the derivative $$f'(x)=2(x-c)g(x)+(x-c)^2g'(x). $$ Then $f'(c)=0$. In fact, if $f$ is divisible by $(x-c)^n$ for $n>1$, $f'(c)=0$.