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##