Descartes' rule of signs

Descartes' Rule of Signs is a theorem in algebra that provides a method to determine the number of positive and negative real roots of a polynomial. Named after the French philosopher and mathematician René Descartes, who introduced the rule in 1637, it serves as a valuable tool in mathematical analysis and its applications.

Overview

Descartes' Rule of Signs states that the number of positive real roots of a polynomial equation can be determined by the number of sign changes in the sequence of its coefficients. Specifically, if the terms of a polynomial are arranged in descending order of their degrees, the number of positive real roots is either equal to the number of sign changes or less than it by an even number. Similarly, to find the number of negative real roots, the rule applies to the polynomial with each variable replaced by its negative counterpart.

Mathematical Formulation

Consider a polynomial \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are real numbers and \(a_n \neq 0\). The rule can be summarized as follows:

1. The number of positive real roots of \(P(x)\) is at most the number of sign changes in the sequence \(a_n, a_{n-1}, \ldots, a_1, a_0\). 2. The number of negative real roots of \(P(x)\) is at most the number of sign changes in the sequence of coefficients when \(x\) is replaced by \(-x\).

Application

Descartes' Rule of Signs is particularly useful in precalculus and calculus for estimating the roots of polynomial equations. It is often employed in conjunction with other methods, such as the Rational Root Theorem and synthetic division, to narrow down the possible real roots of a polynomial.

Limitations

While Descartes' Rule of Signs is a powerful tool, it has limitations. It does not provide information about the exact number of real roots if this number is less than the number of sign changes by an even number. Additionally, it does not account for complex roots, nor does it indicate the multiplicity of roots.

Examples

Consider the polynomial \(P(x) = x^3 - 6x^2 + 11x - 6\). The sequence of coefficients is \(1, -6, 11, -6\), which has two sign changes. Therefore, according to Descartes' Rule of Signs, \(P(x)\) has at most two positive real roots. When considering \(P(-x) = -x^3 - 6x^2 - 11x - 6\), the sequence of coefficients has one sign change, indicating at most one negative real root.

See Also

• Polynomial
• Algebra
• René Descartes
• Rational Root Theorem