Descartes' rule of signs

In mathematics, 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 real roots of a polynomial.

The rule gives us an upper bound number of positive or negative roots of a polynomial. It is not a complete criterion, i.e. it does not tell the exact number of positive or negative roots.

Descartes' rule of signs

Positive roots

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 is less than it by a multiple of 2. Multiple roots of the same value are counted separately.

Negative roots

As a corollary of the rule, the number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by −1, or fewer than it by a multiple of 2. This procedure is equivalent to substituting the negation of the variable for the variable itself: For example, to find the number of negative roots of f(x) = ax³ + bx² + cx + d, we equivalently ask how many positive roots there are for − x in $f(-x)=a(-x)^3+b(-x)^2+c(-x)+d = -ax^3+bx^2-cx+d \equiv g(x).$ Using Descartes' rule of signs on g(x) gives the number of positive roots x_i of g, and since g(x) = f( − x) it gives the number of positive roots ( − x_i) of f, which is the same as the number of negative roots x_i of f.

Example

The polynomial

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

has one sign change between the second and third terms (the sequence of pairs of successive signs is ++, +−, −−). Therefore it has exactly one positive root. Note that the leading sign needs to be considered although in this particular example it does not affect the answer. To find the number of negative roots, change the signs of the coefficients of the terms with odd exponents, to obtain a second polynomial

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

This polynomial has two sign changes (the sequence of pairs of successive signs is −+, ++, +−), meaning that this second polynomial has two or zero positive roots; thus the original polynomial has two or zero negative roots.

In fact, the factorization of the first polynomial is

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

so the roots are −1 (twice) and 1.

The factorization of the second polynomial is

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

So here, the roots are 1 (twice) and −1, the negation of the roots of the original polynomial.

Complex roots

Since any n^th degree polynomial has exactly n roots, the minimum number of complex roots is equal to

$n-(p+q),\,$

where p denotes the maximum number of positive roots, q denotes the maximum number of negative roots (both of which can be found using Descartes' rule of signs), and n denotes the degree of the equation. A simple example is the polynomial

x² + b.

If b > 0, this has no sign changes, and the polynomial does not change when odd-powered terms (of which there are none in this example) have their coefficients multiplied by −1. Thus the maximum number of positive roots is zero, as is the maximum number of negative roots; so the minimum (and in this case exact) number of complex roots is n − (p + q) = 2 − (0 + 0) = 2.

Special case

The subtraction of only multiples of 2 from the maximal number of positive roots occurs because the polynomial may have complex roots, which always come in pairs since the rule applies to polynomials whose coefficients are real. Thus if the polynomial is known to have all real roots, this rule allows one to find the exact number of positive and negative roots. Since it is easy to determine the multiplicity of zero as a root, the sign of all roots can be determined in this case.

Generalizations

If the real polynomial P has k real positive roots counted with multiplicity, then for every a > 0 there are at least k changes of sign in the sequence of coefficients of the Taylor series of the function e^axP(x).

In the 1970s Askold Georgevich Khovanskiǐ developed the theory of fewnomials that generalises Descartes' rule. The rule of signs can be thought of as stating that the number of real roots of a polynomial is dependent on the polynomial's complexity, and that this complexity is proportional to the number of monomials it has, not its degree. Khovanskiǐ showed that this holds true not just for polynomials but for algebraic combinations of many transcendental functions, the so-called Pfaffian functions.

See also

• Sturm's theorem
• Rational root theorem
• Polynomial function theorems for zeros
• Properties of polynomial roots
• Gauss–Lucas theorem

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 Creative Commons Attribution/Share-Alike License.