Polynomials and Derivatives

I am doing an exercise and came to a point where I'd need to know this:

As a consequence of Rolle's theorem, the derivative of a function has a zero whenever our function has more than one zero. But can we say that if all the roots of a polynomial, $p(n)$, are real, all the roots of $p'(n)$ are real?

Yes. Suppose we have a polynomial $p(x)$ of degree $d$, with all roots real (meaning it has $d$ real roots counting multiplicity). If $p(x)$ has a root of multiplicity $m$ at some point $x_0$, then $p'(x)$ has a root of multiplicity $m-1$ at $x_0$, as can be seen by applying product rule: $$\frac{d}{dx}(x-r)^mq(x)=(x-r)^{m-1}(mq(x)+(x-r)q'(x))$$ Also, between any pair of distinct roots of $p(x)$ there must be a root of $p'(x)$ by Rolle's theorem. Thus the number of real roots of $p'(x)$, counting multiplicity, is the number of distinct roots of $p(x)$ minus 1, plus $m-1$ roots for each repeated root of multiplicity $m$, giving a total of $d-1$ (since the number of distinct roots of $p(x)$ plus $m-1$ roots for each repeated root of multiplicity $m$ is the total number of roots of $p(x)$, which is $d$). This is the total number of roots of $p'(x)$, so all its roots must be real.

If all zeros of $p$ lie in the interval $[a,b]$, then all zeros of $p'$ lie in $[a,b]$ by the Gauss–Lucas theorem. More generally, each zero of $p'$ lies in the convex hull of the set of zeros of $p$.