real analysis - Relation between roots of function and roots of its derivative

I'm reading a section my Calculus book that is about the relation between the roots of a polynomial function and the roots of its derivative. So:

Notice that if $x_1$ and $x_2$ are roots of $f$, so that $f(x_1) = f(x_2) = 0$, then by Rolle's Theorem there is a number $x$ between $x_1$ and $x_2$ such that $f'(x) = 0$.

Ok that makes sense. Then:

This means that if $f$ has $k$ different roots $x_1 < x_2 < ... < x_k$, then $f'$ has at least least $k-1$ roots: one between $x_1$ and $x_2$, one between $x_2$ and $x_3$, etc.

That also makes sense, but what confuses me is "at least least $k-1$ roots". Why "at least"? Didn't we just show that there are exactly $k-1$ roots for the derivative, or so to say, if we have a polynomial of degree $n$, then its derivative has $n-1$ roots?

Answer

If $p(x)=x^2+1$, then $p(x)$ has zero roots. However, $p'(x)=2x$, so $p'(x)$ has one root $x=0$.

That argument from your Calculus textbook proves that between any two roots of the original polynomial $p(x)$ there is at least one root of $p'(x)$ between them, but there may be other roots.