If a polynomial and its derivative have the same root, then the root is a repeated root?

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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.
 

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  • #3
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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.
 
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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##
 
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