Descartes’ Rule of Signs

Definition

Descartes’ Rule of Signs is a method used to determine the number of positive and negative real zeros in a polynomial function. It analyzes the sign changes in the coefficients of the polynomial.

5 Must Know Facts For Your Next Test

1. The number of positive real zeros is determined by counting the sign changes in the polynomial's coefficients.
2. The number of negative real zeros is found by applying Descartes' Rule to \( P(-x) \), where \( P(x) \) is the original polynomial.
3. Each sign change indicates a possible zero, but the actual number of zeros could be less by an even number.
4. Descartes’ Rule does not provide exact values for the zeros or their multiplicities, only their possible counts.
5. The rule applies to polynomials with real coefficients and helps to narrow down potential candidates for real zeros.

Review Questions

Related terms

Polynomial:

A mathematical expression consisting of variables, coefficients, and non-negative integer exponents combined using addition, subtraction, and multiplication.

Zero (Root): A solution to the equation \( P(x) = 0 \), where \( P(x) \) is a polynomial function.

$P(-x)$: $P(-x)$ represents substituting $-x$ into a polynomial $P(x)$, used in Descartes’ Rule to find negative real zeros.