Descartes' rule of signs, first published by Renée Descartes in 1637, is a method for determining the largest number of positive or negative roots that a polynomial equation may have. If the terms of the polynomial p(x) are written in the customary fashion--that is, with the terms given in decreasing order of the exponent of x--then the number of positive roots of the polynomial cannot be greater than the number of sign changes in the coefficients. (A sign change occurs whenever one coefficient is positive and the next one is negative, or vice versa. If any coefficients are zero, they are simply ignored.) A test for the number of negative roots can be created by replacing x with -x and counting the sign changes in the new polynomial.

A few examples make the application of the rule easy...