# Descartes' Rule of Signs Descartes' Rule of Signs is a rule in algebra that constrains the number of real roots of a [[Polynomials|polynomial]] with real coefficients, based on the number of sign changes of the polynomial's coefficients. It states that **if $s$ is the number of sign changes (0-coefficients are ignored), and $p$ is the number of the polynomial's positive roots, then $s-p$ must be a positive even number (or 0). The number of negative roots can be similarly constrained by plugging in $-x$ for $x$**. ## Example For example, in the polynomial $f(x) = x^3 - 3x - 2$ there is one sign change so $s=1$, and as $s-p$ must be a positive even number so the number of positive roots must be 1. The number of negative roots can be constrained by plugging in $-x$, so $f(-x) = (-x)^3 - 3(-x) - 2 = -x^3 + 3x -x$. This new polynomial has 2 sign changes, and as $s-p$ must be a positive even number, the number of negative roots is either 0 or 2[^1]. We can't tell which. Indeed, $x^3 - 3x - 2 = (x+1)^2(x-2)$, so there are two positive and one negative root.