Consider a polynomial (we'll look only at third order case, but it easily generalizes) with distinct roots at a, b, and c:

Clearly this function is equal to zero at its roots (s=a, s=b, and s=c). But if we examine its derivative

,

we find that it is not equal to zero at any of the roots.

Now consider a polynomial where the first root is a double root (i.e., it is repeated once):

This function is also equal to zero at its roots (s=a, s=b). If we examine its first derivative

we find that it is still equal to zero at the repeated root (s=a).

In general if we have a polynomial with a root repeated *n*
times, its first *n* derivatives will be equal to zero.

© Copyright 2005 to 2015 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College