Descartes' Rule of Signs Calculator
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Enter Coefficients:
x⁴
x³
x²
x
constant
Results
How to Use
- Enter the coefficients of your polynomial
- Add more terms if needed using the "Add Term" button
- Click "Analyze Polynomial" to apply Descartes' Rule of Signs
- View the analysis of possible real roots
About Descartes' Rule of Signs
Descartes' Rule of Signs states that:
- The number of positive real roots is either equal to the number of sign changes in the polynomial or less than it by an even number
- The number of negative real roots is either equal to the number of sign changes in P(-x) or less than it by an even number
Important notes:
- Zero coefficients are ignored when counting sign changes
- The rule gives an upper bound on the number of real roots
- The actual number of roots may be less than the maximum
- Complex roots are not counted
Common Examples
| Polynomial | Sign Changes | Possible Positive Roots | Possible Negative Roots |
|---|---|---|---|
| x³ - 2x² - x + 2 | 2 | 2 or 0 | 1 |
| x⁴ - 5x² + 4 | 2 | 2 or 0 | 2 or 0 |
| x³ + x² - x - 1 | 1 | 1 | 2 or 0 |
| x⁵ - x⁴ + x³ - x² + x - 1 | 5 | 5, 3, or 1 | 0 |
Applications
- Polynomial root analysis
- Algebraic equation solving
- Mathematical modeling
- Engineering calculations
- Scientific research
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