Descartes' Rule of Signs - Expii

Descartes' rule of signs says that the number of positive real roots of a polynomial (including repeated roots) is less than the number of "sign changes" of the polynomial (for instance, x^5 - x^3 + x^2 + x - 1 has three sign changes, since x^5 is like +x^5). It actually says a little more: the number of positive real roots differs from the number of sign changes by an even number.