In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Descartes' rule of signs, described by
René Descartes
René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
in his ''
La Géométrie
''La Géométrie'' () was published in 1637 as an appendix to ''Discours de la méthode'' ('' Discourse on the Method''), written by René Descartes. In the ''Discourse'', Descartes presents his method for obtaining clarity on any subject. ''La ...
'', counts the
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusin ...
of a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
by examining sign changes in its coefficients. 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 particular, when the number of sign changes is zero or one, then there are exactly zero or one positive roots.
A
linear fractional transformation
In mathematics, a linear fractional transformation is, roughly speaking, an inverse function, invertible transformation of the form
: z \mapsto \frac .
The precise definition depends on the nature of , and . In other words, a linear fractional t ...
of the variable makes it possible to use the rule of signs to count roots in any interval. This is the basic idea of
Budan's theorem
In mathematics, Budan's theorem is a theorem for bounding the number of real roots of a polynomial in an interval, and computing the parity of this number. It was published in 1807 by François Budan de Boislaurent.
A similar theorem was publish ...
and the
Budan–Fourier theorem. Repeated division of an interval in two results in a set of disjoint intervals, each containing one root, and together listing all the roots. This approach is used in the fastest algorithms today for computer computation of real roots of polynomials (see
real-root isolation
In mathematics, and, more specifically in numerical analysis and computer algebra, real-root isolation of a polynomial consist of producing disjoint intervals of the real line, which contain each one (and only one) real root of the polynomial, and ...
).
Descartes himself used the transformation for using his rule for getting information of the number of negative roots.
Descartes' rule of signs
Positive roots
The rule states that if the nonzero terms of a single-variable
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
with
real coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s are ordered by descending variable exponent, then the number of positive
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusin ...
of the polynomial is either equal to the number of sign changes between consecutive (nonzero) coefficients, or is less than it by an even number. A root of
multiplicity is counted as roots.
In particular, if the number of sign changes is zero or one, the number of positive roots equals the number of sign changes.
Negative roots
As a
corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
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 an even number. This procedure is equivalent to substituting the negation of the variable for the variable itself.
For example, the negative roots of
are the positive roots of
:
Thus, applying Descartes' rule of signs to this polynomial gives the maximum number of negative roots of the original polynomial.
Example: cubic polynomial
The polynomial
:
has one sign change between the second and third terms, as the sequence of signs is . Therefore, it has exactly one positive root.
To find the number of negative roots, change the signs of the coefficients of the terms with odd exponents, i.e., apply Descartes' rule of signs to the polynomial
This polynomial has two sign changes, as the sequence of 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
In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...
of the first polynomial is
:
so the roots are −1 (twice) and +1 (once).
The factorization of the second polynomial is
:
So here, the roots are +1 (twice) and −1 (once), the negation of the roots of the original polynomial.
Proof
The following is a rough outline of a proof. First, some preliminary definitions:
* Write the polynomial
as
where we have integer powers
, and nonzero coefficients
.
* Let
be the number of sign changes of the coefficients of
, meaning the number of
such that
.
* Let
be the number of strictly positive roots (counting multiplicity).
With these, we can formally state Descartes' rule as follows:
If
, then we can divide the polynomial by
, which would not change its number of strictly positive roots. Thus WLOG, let
.
Nonreal roots
Any ''n''th degree polynomial has exactly ''n'' roots in the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
, if counted according to multiplicity. So if ''f''(''x'') is a polynomial with real coefficients which does not have a root at 0 (that is a polynomial with a nonzero constant term) then the
minimum number of nonreal roots is equal to
:
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 polynomial.
Example: some zero coefficients and nonreal roots
The polynomial
:
has one sign change; so the maximum number of positive real roots is one. As
:
has no sign change, the original polynomial has no negative real roots. So the minimum number of nonreal roots is
:
Since nonreal roots of a polynomial with real coefficients must occur in conjugate pairs, it means that has exactly two nonreal roots and one real root, which is positive.
Special case
The
even difference between the number of sign changes and the number of positive roots is positive when the polynomial has pairs of
conjugate nonreal roots with positive
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. Thus, if the polynomial is known to have all real roots, Descartes' 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
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of the function ''e''
''ax''''P''(''x''). For sufficiently large ''a'', there are exactly ''k'' such changes of sign.
In the 1970s
Askold Khovanskii 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 function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction ...
s, the so-called
Pfaffian functions.
See also
*
*
*
*
Notes
External links
{{PlanetMath attribution, id=5997, title=Descartes' rule of signs
Descartes' Rule of Signs– Proof of the rule
– Basic explanation
Theorems about polynomials