Proof: If a Polynomial & its Derivative have Same Root

[Happiness]

Given a polynomial $f(x)$. Suppose there exists a value $c$ such that $f(c)=f'(c)=0$, where $f'$ denotes the derivative of $f$. Then $f(x)=(x-c)^mh(x)$, where $m$ is an integer greater than 1 and $h(x)$ is a polynomial.

Is it true? Could you prove it?

Note: The converse is true and can be proved easily.

 

[mfb]

Looks quite trivial to me.

 

[Happiness]

Looks quite trivial to me.

It's trivial if we know what to use.

I managed to prove it using factor theorem. But I guessed I wanted to see if there is a more elementary proof when I posted the question. And then I found an elementary proof for factor theorem.

 

[PeroK]

It's trivial if we know what to use.

I managed to prove it using factor theorem. But I guessed I wanted to see if there is a more elementary proof when I posted the question. And then I found an elementary proof for factor theorem.

If $f(c) = 0$ then $f(x) = (x-c)g(x)$ and $f'(x) = g(x) + (x-c)g'(x) \dots$