Search results
Results From The WOW.Com Content Network
Descartes' rule of signs In mathematics, Descartes' rule of signs, described by René Descartes in his La Géométrie, counts the roots of a polynomial 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 polynomial's coefficients (omitting zero coefficients), and the difference between the root count and the ...
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 Géométrie and two other appendices, also by Descartes, La Dioptrique (Optics) and Les Météores (Meteorology), were published with the Discourse to give examples of the ...
In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who stated it in 1643.
Descartes' rule of signs All results described in this article are based on Descartes' rule of signs. If p(x) is a univariate polynomial with real coefficients, let us denote by #+(p) the number of its positive real roots, counted with their multiplicity, [1] and by v(p) the number of sign variations in the sequence of its coefficients.
Therefore, they require to start with an interval such that the function takes opposite signs at the end points of the interval. However, in the case of polynomials there are other methods (Descartes' rule of signs, Budan's theorem and Sturm's theorem) for getting information on the number of roots in an interval.
Descartes' rule of signs and its generalizations Descartes' rule of signs asserts that the difference between the number of sign variations in the sequence of the coefficients of a polynomial and the number of its positive real roots is a nonnegative even integer.
Regulae ad directionem ingenii, or Rules for the Direction of the Mind is an unfinished treatise regarding the proper method for scientific and philosophical thinking by René Descartes. Descartes started writing the work in 1628, and it was eventually published in 1701 after Descartes' death. [1] This treatise outlined the basis for his later work on complex problems of mathematics, geometry ...
See also Descartes' rule of signs – Counting polynomial real roots based on coefficients Marden's theorem – On zeros of derivatives of cubic polynomials Newton's identities – Relations between power sums and elementary symmetric functions Quadratic function#Upper bound on the magnitude of the roots