Descartes Rule of Signs Calculator

Descartes’ Rule of Signs gives an upper bound on the number of positive and negative real roots of a polynomial — no computation needed, just counting sign changes in the coefficients.

For positive roots: write out the nonzero coefficients in order from highest to lowest degree. Count how many times the sign changes from one term to the next. The number of positive real roots equals this count, or that count minus 2, or minus 4, and so on down to 0 or 1.

For negative roots: replace x with -x in the polynomial (which flips the sign of every odd-degree term), then count sign changes in the new coefficients. The number of negative real roots follows the same rule.

Example: f(x) = x³ - 2x - 1. Coefficients: +1, 0, -2, -1. Ignoring the zero, the sequence is +, -, -, which changes sign once. One sign change means exactly one positive real root: the count can only drop by 2 at a time, and there is nothing below 1 to drop to.

Now substitute -x. f(-x) = -x³ + 2x - 1, with coefficients -1, 0, +2, -1. That sequence is -, +, -, so it changes sign twice, and there are either 2 negative real roots or none.

It is worth being clear about where the rule stops. One positive root out of degree 3 leaves two roots unaccounted for, and they are either both negative or a complex conjugate pair. Both readings fit everything counted so far, so Descartes cannot choose between them. Factoring settles it: x³ - 2x - 1 = (x + 1)(x² - x - 1), and the quadratic gives (1 ± √5)/2, which is 1.618 and -0.618. So the roots are -1, -0.618 and 1.618, and the answer was 2 negative. The rule narrowed three possibilities to two. Finishing the job was never its work.

Zero coefficients are skipped when counting sign changes. They do not interrupt the sequence.

The rule is useful as a quick filter. Before solving a polynomial numerically, knowing “at most 2 positive roots” saves you from looking in the wrong places. It dates to Descartes’ La Geometrie of 1637 — nearly 400 years of practical use.