Iterated Rolle

Induction on the number of derivatives. Let $r_0=n$ be the number of distinct real roots of $P^{(0)}=P$. We prove that for every $k\in\{0,1,\ldots,n-1\}$, $P^{(k)}$ has at least $n-k$ distinct real roots.

Base case. $k=0$: $P$ has $n$ distinct real roots by hypothesis, and $n-0=n$.

Inductive step. Suppose $P^{(k)}$ has at least $n-k$ distinct real roots, call them $x_1<x_2<\cdots<x_{n-k}$. Since $P^{(k)}$ is differentiable on each interval $]x_i,x_{i+1}[$ and continuous on $[x_i,x_{i+1}]$, and $P^{(k)}(x_i)=P^{(k)}(x_{i+1})=0$, Rolle's theorem gives a point $c_i\in]x_i,x_{i+1}[$ with
$$\bigl(P^{(k)}\bigr)'(c_i)=P^{(k+1)}(c_i)=0.$$
There are $n-k-1$ such intervals, so $P^{(k+1)}$ has at least $n-k-1$ distinct real roots. This completes the induction.

Conclusion. Taking $k=n-1$, $P^{(n-1)}$ has at least $n-(n-1)=1$ real root.