Search results
Results From The WOW.Com Content Network
Anderson's theorem (real analysis) Andreotti–Frankel theorem (algebraic geometry) Angle bisector theorem (Euclidean geometry) Ankeny–Artin–Chowla theorem (number theory) Anne's theorem ; Apéry's theorem (number theory) Apollonius's theorem (plane geometry) Appell–Humbert theorem (complex manifold) Arakelyan's theorem (complex analysis)
is a factor of P(x) with real coefficients. Repeating this for all non-real factors gives a factorization with linear or quadratic real factors. For computing these real or complex factorizations, one needs the roots of the polynomial, which may not be computed exactly, and only approximated using root-finding algorithms.
If two or more factors of a polynomial are identical, then the polynomial is a multiple of the square of this factor. The multiple factor is also a factor of the polynomial's derivative (with respect to any of the variables, if several). For univariate polynomials, multiple factors are equivalent to multiple roots (over a suitable extension field).
The theorem is a special case of the polynomial remainder theorem. [1] [2] The theorem results from basic properties of addition and multiplication. It follows that the theorem holds also when the coefficients and the element belong to any commutative ring, and not just a field.
for non-zero real constants A, B, C, ω, this function is well-defined for all (t, x, y, z), but it cannot be solved explicitly for these variables and written as "t =", "x =", etc. The implicit function theorem of more than two real variables deals with the continuity and differentiability of the function, as follows. [4]
This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be an "example" as the term should be understood here (an elegant proof of an isolated but particularly striking fact, as opposed to a proof of a ...
It is clear that any finite set {} of points in the complex plane has an associated polynomial = whose zeroes are precisely at the points of that set. The converse is a consequence of the fundamental theorem of algebra: any polynomial function () in the complex plane has a factorization = (), where a is a non-zero constant and {} is the set of zeroes of ().
Littlewood's three principles are quoted in several real analysis texts, for example Royden, [2] Bressoud, [3] and Stein & Shakarchi. [4] Royden [5] gives the bounded convergence theorem as an application of the third principle. The theorem states that if a uniformly bounded sequence of functions converges pointwise, then their integrals on a ...