The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

The Most Marvelous Theorem in Mathematics, Dan Kalman

As stated on the previous page, for the polynomial `a`_{n} `z`^{n} + `a`_{n − 1} `z`^{n − 1} + ··· + `a`_{1} `z` + `a`_{0}, the sum of the roots is given by −`a`_{n − 1}/`a`_{n}. Therefore the *average* of the roots is given by −`a`_{n − 1}/(`n``a`_{n}). Now the derivative of this polynomial is `n``a`_{n} `z`^{n − 1} + (`n` − 1)`a`_{n − 1} `z`^{n − 2} + ··· + `a`_{1}. 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.