Apply Descartes' Rule of Signs to any polynomial. Count sign changes in f(x) and f(−x) to determine the possible number of positive, negative, and complex roots.
Descartes' Rule of Signs is a powerful theorem in algebra that tells you the possible number of positive and negative real roots of a polynomial without actually solving it. The rule states that the number of positive real roots of a polynomial f(x) is either equal to the number of sign changes in the sequence of its coefficients, or less than that by an even number. Similarly, the number of negative real roots equals the number of sign changes in f(−x), or less by an even number. Any remaining roots must be complex.
This calculator lets you enter the coefficients of any polynomial and instantly see the sign-change analysis for both f(x) and f(−x). It highlights every sign change in the coefficient sequence, lists all possible distributions of positive, negative, and complex roots, and presents the results in a visual stacked-bar format. Two input modes are available: comma-separated coefficients for quick entry, or individual fields for each degree. Presets for classic polynomials let you explore the rule immediately. Whether you are narrowing down root possibilities before applying the Rational Root Theorem, checking your factoring work, or studying polynomial behavior in an algebra or precalculus course, Descartes' Rule gives you a fast, reliable upper bound on the number of roots of each type.
descartes-rule-of-signs helps you solve descartes-rule-of-signs problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Coefficients (highest degree first), constant once and immediately inspect Polynomial f(x), Sign changes in f(x), f(−x) to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Descartes' Rule: The number of positive real roots of f(x) equals the number of sign changes in the coefficient sequence, or less by a positive even integer. For negative roots, apply the same rule to f(−x). Complex roots account for the remainder.
Result: Polynomial f(x) shown by the calculator
Using the preset "x³−6x²+11x−6", the calculator evaluates the descartes-rule-of-signs setup, applies the selected algebra rules, and reports Polynomial f(x) with supporting checks so you can verify each transformation.
This calculator takes Coefficients (highest degree first), constant and applies the relevant descartes-rule-of-signs relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Polynomial f(x), Sign changes in f(x), f(−x), Sign changes in f(−x) to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
It tells you the maximum possible number of positive and negative real roots of a polynomial, and that the actual count differs from the maximum by an even number. Any remaining roots are complex.
No. It gives a list of possibilities. For example, 3 sign changes means 3 or 1 positive roots. You need other methods (Rational Root Theorem, graphing, or numerical solving) to determine the exact count.
Zero coefficients are skipped. A sign change is counted only between consecutive nonzero coefficients.
Because complex roots of a real polynomial come in conjugate pairs, removing 2 real roots at a time and replacing them with 2 complex roots. So the count drops by 2, 4, etc.
No. It only counts possible numbers of positive and negative roots. To find the actual root values, use factoring, the Rational Root Theorem, synthetic division, or numerical methods.
No. Descartes' Rule of Signs applies only to polynomials with real coefficients. The notion of 'sign' is not defined for complex numbers.