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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 ...
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.
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.
La Géométrie. The work was the first to propose the idea of uniting algebra and geometry into a single subject [2] and invented an algebraic geometry called analytic geometry, which involves reducing geometry to a form of arithmetic and algebra and translating geometric shapes into algebraic equations. For its time this was ground-breaking.
Descartes' work provided the basis for the calculus developed by Leibniz and Newton, who applied the infinitesimal calculus to the tangent line problem, thus permitting the evolution of that branch of modern mathematics. [139] His rule of signs is also a commonly used method to determine the number of positive and negative roots of a polynomial.
Cartesian circle. The Cartesian circle (also known as Arnauld 's circle[1]) is an example of fallacious circular reasoning attributed to French philosopher René Descartes. He argued that the existence of God is proven by reliable perception, which is itself guaranteed by God.
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
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. It results that if this number of sign variations is zero, then the polynomial does not have any positive real roots, and, if this ...