The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman

Average of roots equals average of roots of derivative

As stated on the previous page, for the polynomial an zn + an − 1 zn − 1 + ··· + a1 z + a0, the sum of the roots is given by an − 1/an. Therefore the average of the roots is given by an − 1/(nan). Now the derivative of this polynomial is nan zn − 1 + (n − 1)an − 1 zn − 2 + ··· + a1. Arguing as before, the average of the roots of the derivative is

This shows that the average of the roots of the derivative is equal to the average of the roots of the original polynomial.