Descartes Rule Calculator

Descartes Rule Calculator

Count sign changes to estimate the possible number of positive and negative real roots.

Last updated: June 2026 | By Patchworkr Team

Polynomial analysis

Build your expression

Use `^` for exponents. Invalid terms are rejected.

Summary
Standard form
x^4 -2x^3 +3x^2 -4x +5 = 0
Degree
4
Total roots
4
Positive roots
Sign changes4
420
Negative roots
Sign changes in P(-x)0
0

What the rule says

The number of positive real roots is the count of sign changes in P(x), or less than that by an even number.

For negative roots, apply the same idea to P(-x). Zero coefficients are ignored when counting sign changes.

Example

x^4 - 2x^3 + 3x^2 - 4x + 5

This example has 4 sign changes for positive roots and 0 sign changes for negative roots.

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