Descartes’ Rule of Signs Calculator
This calculator applies Descartes’ Rule of Signs to estimate the number of positive and negative real roots of a polynomial equation. It’s a powerful tool in algebra for analyzing root behavior without solving the equation.
Analyze Positive and Negative Real Roots of Polynomials
Descartes’ Rule of Signs Explanation
Explanation:
- Count the number of sign changes in the coefficients of $$f(x)$$ for possible positive roots.
- Replace $$x$$ with $$-x$$ and count sign changes in $$f(-x)$$ for possible negative roots.
- The actual number of roots is equal to or less than the number of sign changes, differing by an even number.
Descartes’ Rule – Calculation Example
Polynomial: f(x) = x⁴ – 3x³ + 2x² – x + 6
Step 1: Coefficients signs: + – + – + → 4 sign changes
→ Up to 4, 2, or 0 positive real roots
Step 2: f(−x) = x⁴ + 3x³ + 2x² + x + 6
Signs: + + + + + → 0 sign changes
→ 0 negative real roots
Result: Up to 4 positive real roots, 0 negative real roots
Descartes’ Rule of Signs is a classic method in algebra for root estimation, especially useful when factoring high-degree polynomials. While it doesn’t provide exact root values, it narrows down the possibilities and helps guide further solving methods like synthetic division or graphing.