Descartes' Rule of Signs

Determine the possible number of positive and negative real roots of a polynomial equation.

Last updated: March 2026 | By ForgeCalc Engineering

Enter Polynomial

Use ^ for exponents (e.g., x^2 for x squared)

Polynomial Analysis

Standard Form: x^4 - 2x^3 + 3x^2 - 4x + 5 = 0

Degree: 4

Total roots (including complex): 4

Positive Real Roots

Sign changes:4

Possible number of positive roots:

420

Negative Real Roots

Sign changes in P(-x):0

Possible number of negative roots:

Remaining roots are complex conjugate pairs: Varies based on actual positive/negative root count

What is Descartes' Rule of Signs?

Descartes' Rule of Signs is a technique for determining the possible number of positive and negative real roots of a polynomial equation. It works by counting sign changes in the coefficients.

The rule states that the number of positive real roots is either equal to the number of sign changes in P(x), or less than that by an even number. The same applies to negative roots using P(-x).

How to Apply the Rule

For Positive Roots

Write polynomial in standard form (descending powers)
Count the number of sign changes between consecutive non-zero coefficients
The number of positive roots equals this count, or less by an even number

For Negative Roots

Replace x with -x in the polynomial: P(-x)
Count sign changes in the resulting polynomial
The number of negative roots equals this count, or less by an even number

Important Notes

The rule gives possible numbers, not the exact count
Complex roots always come in conjugate pairs
The total number of roots (including complex) equals the degree
Zero coefficients are ignored when counting sign changes

Example Analysis

Analyze: x⁴ - 2x³ + 3x² - 4x + 5 = 0

Positive Roots:

Coefficients: +1, -2, +3, -4, +5

Sign changes: + to -, - to +, + to -, - to + = 4 changes

Possible positive roots: 4, 2, or 0

Negative Roots:

P(-x) = x⁴ + 2x³ + 3x² + 4x + 5

All coefficients positive = 0 sign changes

Possible negative roots: 0

Conclusion: 0 to 4 positive real roots, 0 negative real roots, and complex conjugate pairs for the rest.

Frequently Asked Questions

Does this rule give exact root counts?

No, it gives possible counts. The actual number of positive (or negative) real roots equals the sign changes, or is less by an even number (2, 4, 6, etc.).

What if a coefficient is zero?

Skip it! Only count sign changes between non-zero coefficients. For example, in x³ + 0x² - 1, there's only one sign change (+ to -).

Can this tell me the actual roots?

No, this rule only tells you how many positive and negative real roots are possible. You need other methods (factoring, graphing, numerical methods) to find the actual roots.