If $(x-r)$ is a repeating factor of the polynomial $f(x)=0$ repeating $m$ times, how many times does it repeat for $f'(x)$?

Question

This is the question in my book: if $x-r$ is a factor of the polynomial $$ f(x) = a_{n}x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \ldots + a_0 $$ repeated $m$ times, then $r$ is a root of $f'(x)=0$ repeated how many times?

This is the answer given: $x=r$ is root of $f'(x)=0$ repeated $(m-1)$ times. How did they reach this conclusion? Is there some theorem that explains this?

Answer 1

$$f(x)=g(x)h(x)$$ $$g(x)=(x-r)^m\rightarrow g'(x)=m(x-r)^{m-1}$$ Product rule: $$f'(x)=(x-r)^mh'(x)+m(x-r)^{m-1}h(x)$$ $$\rightarrow f'(x)=(x-r)^{m-1}[(x-r)h'(x)+mh(x)]$$ So $x=r$ is the root of $f'(x)$ $m-1$ times as the question states