Statement about Rolle's Theorem (true or false?) There's a statement, that I believe is false 

Between two distinct zeroes of a polynomial $p$, there is a number $c$ such that $p′(c) = 0$.

Here is my reasoning: 


*

*A polynomial of an even degree has a derivative of an odd degree, so it has no root, in this case the theorem fails.

*The statement doesn't say that there's at least a number $c$.


Therefore, the statement fails. Is my thinking process correct?
 A: 
There's a statement, that I believe is false
Between two distinct zeroes of a polynomial $p$, there is a number $c$ such that $p′(c) = 0$

I'll begin by saying, a function which satisfies the hypotheses of Rolle's Theorem is guaranteed its conclusions.

A polynomial of an even degree has a derivative of an odd degree, so it has no root, in this case the theorem fails.

A polynomial of even degree indeed has a derivative with odd degree. However, this does not imply the existence (or lack thereof) of a function's real roots.
Take for example the even function $f(x) = x^2$. It has one real root located at $x = 0$. It has an odd derivative $f'(x) = 2x$.
It has an infinite number of intervals $[a, b]$ such that $f(a) = f(b)$, all of which satisfy the hypotheses of Rolle's Theorem.
A: Think about it this way: suppose by contradiction that there is no point between two zeroes of a polynomial s.t the derivative is 0. Since the derivative is continuous, then between the two zeroes of the polynomial the derivative must either be strictly increasing or strictly decreasing. 
However, you clearly cannot have a value $a$ such that $f(a)=0$, increase it for a bit, then get a value $b$ with $f(b)=0$. Thus a contradiction has been reached.
