Descartes Rule of Signs Calculator
Count possible positive and negative real roots of a polynomial using Descartes' Rule of Signs.
Enter coefficients up to degree 5 for sign-change analysis.
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. Sign changes (ignoring zeros): + to - is one change. So there is exactly one positive real root. For negative roots: f(-x) = -x³ - (-2x) - 1… wait. Actually for f(-x) = -x³ + 2x - 1: sign sequence -, +, -: two sign changes. So there are 2 or 0 negative real roots. Combined with the degree-3 constraint and one positive root: there must be 2 negative real roots (since complex roots come in pairs).
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.
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