Let $r_0,r_1,...,r_m$ be the real roots of $a_nx^n+a_{n-1}x^{n-1}+...+a_0$.Is there a closed-form expression for $\sum_{i=1}^mr_i -\sum_{i=1}^m1/r_i$?

Question

Let $r_0, r_1, ... ,r_m$ be the real roots of $a_nx^n+a_{n-1}x^{n-1}+...+a_0$, with $a_0\ne0.$ Is there a closed-form expression for $$ \ \ \ \ \ \sum_{i=1}^mr_i - \sum_{i=1}^m \frac{1}{r_i} \ \ \ \ ?$$

Answer 1

$\sum_{i=1}^mr_i=-\frac{a_{n-1}}{a_n}$. Now substitute $\frac{1}{x}$ for $x$ then $\sum_{i=1}^m \frac{1}{r_i}=-\frac{a_{1}}{a_0}$.