Descartes' Rule of Signs
Descartes' Rule of Signs states that the number of positive real roots of a polynomial p(x) with real coefficients is either equal to the number of sign changes between consecutive nonzero coefficients, or is less than that number by an even integer. Similarly, the number of negative real roots equals the number of sign changes in p(−x), or differs by a positive even integer. This rule provides an upper bound on the number of positive and negative real roots without actually solving the polynomial.
Worked Example
Problem
Apply Descartes' Rule of Signs to p(x) = x⁵ − 3x⁴ + 2x³ + x² − 4x + 1 to determine possible numbers of positive and negative real roots.
Solution
Step 1 (Positive roots): Write coefficients in order: +1, −3, +2, +1, −4, +1. Count sign changes: (+→−)=1, (−→+)=2, (+→+)=none, (+→−)=3, (−→+)=4. Total: 4 sign changes. Possible positive real roots: 4, 2, or 0. Step 2 (Negative roots): Replace x with −x: p(−x) = (−x)⁵ − 3(−x)⁴ + 2(−x)³ + (−x)² − 4(−x) + 1 = −x⁵ − 3x⁴ − 2x³ + x² + 4x + 1. Coefficients: −1, −3, −2, +1, +4, +1. Sign changes: (−→+)=1, one change only. Possible negative real roots: 1. Step 3: The polynomial has degree 5, so with 1 negative root and possibly 4, 2, or 0 positive roots, the remaining roots are complex.
Answer
Possible positive real roots: 4, 2, or 0; Possible negative real roots: 1
Applying Descartes' Rule — Examples with Various Polynomials
| Polynomial | Sign Changes p(x) | Sign Changes p(−x) | Max Pos. Roots | Max Neg. Roots |
|---|---|---|---|---|
| x² − 5x + 6 | 2 | 0 | 2 or 0 | 0 |
| x³ + x + 1 | 0 | 2 | 0 | 2 or 0 |
| x⁴ − x³ + x² − x + 1 | 4 | 0 | 4,2, or 0 | 0 |
| 2x³ − x + 1 | 2 | 1 | 2 or 0 | 1 |
| x² + 4 | 0 | 0 | 0 | 0 |
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Related Terms
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that every non-constant polynomial with complex (including real) coefficients has at least one complex root, and consequently a polynomial of degree n has exactly n roots counted with multiplicity in the complex number system. This theorem guarantees that the complex numbers are algebraically closed, meaning no further number extensions are needed to solve polynomial equations. It was first proved rigorously by Carl Friedrich Gauss in his 1799 doctoral dissertation and is one of the most important results in all of mathematics.
Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the highest exponent n is called the degree. Polynomials are used extensively in calculus, numerical analysis, and computer science for approximating functions and solving complex problems.
This rule is named after René Descartes (1596–1650), the French mathematician and philosopher who stated it (without proof) in his 1637 appendix "La Géométrie," part of his philosophical work "Discours de la Méthode." A rigorous proof was provided later by other mathematicians, including Gauss.