Root average of polynomials under differentiation.

Question

Let $$ P(x)= a_n x^n+a_{n-1} x^{n-1} + \cdots+a_1 x + a_0$$ be a polynomial with real coefficients and $n\ge 2$.

Prove that the average of roots of $P(x)$ is invariant under differentiation.

Answer 1

Average of roots of $P(x)$ is $$-\frac{a_{n-1}}{na_n}$$

Average of roots of $P'(x)$ is $$-\frac{(n-1)a_{n-1}}{n(n-1)a_n}=-\frac{a_{n-1}}{na_n}$$

Since it is true for an arbitrary polynomial $P(x)$ and it's first derivative $P'(x)$ it is true for all polynomials by induction.