We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$. Show that there exists a real number $\eta > 0$ such that for every $r \in ]1-\eta; 1[$, the polynomial $p(rX)$ is split, has exactly $\sigma(p)$ roots inside the interval $]-1; 1[$ and has no stable root.
We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
Show that there exists a real number $\eta > 0$ such that for every $r \in ]1-\eta; 1[$, the polynomial $p(rX)$ is split, has exactly $\sigma(p)$ roots inside the interval $]-1; 1[$ and has no stable root.