Boyd & Vandenberghe, problem 3.51 — how to show that the number of roots is even?

Question

This question is related to Problem 3.51 of Convex Optimization. Consider a polynomial $p(x)$ which has $n$ roots i.e. $$p(x)=(x-s_1)(x-s_2)\cdots(x-s_n)$$ where we assume w.l.o.g. that $s_1\leq s_2\cdots \leq s_n$. In this case if the polynomial is positive in $$x\in (s_k,s_{k+1})$$ then how to show that the number of roots present on the right side of the interval $(s_k,s_{k+1})$ are even? Any help in this regard will be much appreciated. Thanks in advance.

Answer 1

The sign of $p(x)$ for $x \in (s_k,s_{k+1})$ is obtained by multiplying the numbers $\pm 1$ for each of the factors. Note that $x-s_j>0$ for $j <k$ and $<0$ for $j >k$. If there are an odd number of factors to the right of $(s_k,s_{k+1})$ then the value of $p$ in this interval would be negative.