Descartes' rule of signs
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Descartes' rule of signs, first described by René Descartes in his work La Géométrie, 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
has one sign change between the second and third terms. Therefore it has exactly 1 positive root. In fact, one can see that this polynomial factors as
so the roots are −1 (twice) and 1.
Negating the odd-power terms gives
This polynomial has two sign changes, meaning the original polynomial has 2 or 0 negative roots and this second polynomial has 2 or 0 positive roots. The factorization of the second polynomial is
so here the roots are 1 (twice) and −1, the negation of the roots of the original polynomial.
[edit] Special case
Note that if the polynomial is known to have all real roots, then this rule allows one to find the exact number of positive roots. Since it is easy to determine the multiplicity of zero as a root, this therefore allows the determination of the number of negative roots as well. Thus the sign of all roots can be determined in this case.
[edit] See also
[edit] External Links
- Descartes’ Rule of Signs — Proof of the Rule
This article incorporates material from Descartes' rule of signs on PlanetMath, which is licensed under the GFDL.
