Prove that if p(x) is a polynomial with n distinct zeros then p'(x) has at least n-1 zeros.
It seems that this is a fairly simply use of Rolle's theorem but I can't seem to see how to do it! Help would be great!
Here is an idea on how to get started. To simplify thing lets suppose that has two roots lets call them and with out loss of generality we can have .
Rolle's theorem states that if then for some .
The above statement proves that if has two roots then has roots.
See if you can generalize this argument.