Descartes' Rule of Signs Calculator
Enter polynomial coefficients from highest to lowest degree and instantly see the maximum number of positive and negative real roots — with a step-by-step sign-change analysis for p(x) and p(−x).
Could be 2 or 0 (decreasing by 2)
Coefficients from highest to lowest power
Non-zero sign sequence for p(x)
Sign changes → possible positive real roots
Coefficients of p(−x) — odd-degree terms negated
Non-zero sign sequence for p(−x)
Sign changes → possible negative real roots
How does this calculator work?
Count sign changes in p(x)'s non-zero coefficients for the max positive real roots (or that minus 2, 4, …); apply the same count to p(−x) — where odd-degree coefficients are negated — for negative real roots. Enter comma-separated coefficients from highest to lowest degree.
Formula
How this is calculated
Descartes' rule of signs is a fast way to bound how many real roots a polynomial can have, before solving it. Start with the non-zero coefficients of p(x) written from highest to lowest degree. Count the number of times consecutive non-zero terms change sign (plus to minus, or minus to plus). That count is the maximum number of positive real roots; the actual count is that number or any smaller value obtained by subtracting 2 repeatedly, down to 0 or 1.
For negative real roots, apply the same rule to p(−x). Replacing x with −x negates the coefficient of every odd-degree term and leaves even-degree terms unchanged. Count the sign changes in the resulting sequence — that gives the upper bound on negative real roots, again decreasing by 2 for the possible values.
The rule gives an upper bound and eliminates impossible root counts, but it does not find the roots themselves. It is exact only when all roots are real; complex roots come in conjugate pairs and consume sign changes without being positive or negative reals. Zero is neither counted as positive nor negative, so any zero root reduces the polynomial's effective degree.
Frequently asked questions
A sign change occurs between two consecutive non-zero coefficients when one is positive and the next is negative (or vice versa). Zero coefficients are skipped — they do not count as sign changes or break the sequence.
Complex roots of polynomials with real coefficients come in conjugate pairs (a+bi and a−bi). Each complex pair uses two sign changes without contributing a real root, so the actual positive or negative root count can be the maximum minus 2, 4, 6, and so on.
Not always. The rule gives an upper bound and lists all possible counts, but the actual count depends on the specific polynomial. For example, 2 sign changes means 2 or 0 positive roots — you need further analysis (or a root-finder) to determine which.
TG we-Calculate Editorial Team. (2026). Descartes' Rule of Signs Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/descartes-rule-of-signs-calculator
TG we-Calculate Editorial Team. "Descartes' Rule of Signs Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/descartes-rule-of-signs-calculator.
TG we-Calculate Editorial Team, "Descartes' Rule of Signs Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/descartes-rule-of-signs-calculator
@misc{wecalculate_descartes_rule_of_signs_calculator, title = {Descartes' Rule of Signs Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/descartes-rule-of-signs-calculator}}, year = {2026}, note = {TG we-Calculate} }
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