Descartes' rule of signs

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Descartes' rule of signs, first described by René Descartes in his work La Geometrie, is a technique for determining the number of positive or negative roots of a polynomial.

The rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or less than it by a multiple of 2. Multiple roots of the same value are counted separately.

As a corollary of the rule, the number of negative roots is the number of sign changes after negating the coefficients of odd-power terms (otherwise seen as substituting the negation of the variable for the variable itself), or less than it by a multiple of 2.

For example, the polynomial

x^3 + x^2 - x - 1 \,

has one sign change between the second and third terms. Therefore it has exactly 1 positive root.

Negating the odd-power terms gives

-x^3 + x^2 + x - 1 \,

This polynomial has two sign changes, so the original polynomial has 2 or 0 negative roots.

The polynomial factors easily as

(x + 1)^{2}(x - 1) \,,

so the roots are -1 (twice) and 1.

This article incorporates material from Descartes' rule of signs on PlanetMath, which is licensed under the GFDL.

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