Descartes' Rule of Signs relates the number of positive real roots of a polynomial with real coefficients to the number of sign changes between consecutive nonzero coefficients, stating that the number of positive roots either equals that count or is less than it by an even number, with the same statement applying to negative roots after substituting the negative of the variable. Named for Rene Descartes, it gives a quick bound on a polynomial's real roots without solving the equation.
Facts
StatementThe number of positive real roots of a polynomial with real coefficients either equals the number of sign changes between its consecutive nonzero coefficients or is less than that count by an even number; applying the rule to p(-x) bounds the negative real roots the same way. 1 Classification
Statement Form Statement Form Connections
Has Statement Form
In Branch
Source Descartes' Rule of Signs (Wikipedia)
Proved By
Sources
1. Descartes' Rule of Signs (Wikipedia)
Wikimedia FoundationLead paragraph, second sentence
The number of positive real roots is at most the number of sign changes in the sequence of the polynomial's coefficients (omitting zero coefficients), and the difference between the root count and the sign change count is always even.
In Group: Polynomials and Elementary Algebra, Lead section
René Descartes in his La Géométrie, counts the roots of a polynomial by examining sign changes in its coefficients.
In Branch: Algebra, Lead section
by René Descartes in his La Géométrie, counts the roots of a polynomial by examining sign changes in its coefficients.
View the Source 2. Wikidata: Descartes' Rule of Signs
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