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Fundamental theorem of algebra

The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at leas

Complex conjugate root theorem

In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjug

Hilbert's irreducibility theorem

In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational numb

Eilenberg–Niven theorem

Eilenberg–Niven theorem is a theorem that generalizes the fundamental theorem of algebra to quaternionic polynomials, that is, polynomials with quaternion coefficients and variables. It is due to Samu

Grace–Walsh–Szegő theorem

In mathematics, the Grace–Walsh–Szegő coincidence theorem is a result named after John Hilton Grace, Joseph L. Walsh, and Gábor Szegő.

Polynomial remainder theorem

In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states that the remainder of the divisio

Cohn's theorem

In mathematics, Cohn's theorem states that a nth-degree self-inversive polynomial has as many roots in the open unit disk as the reciprocal polynomial of its derivative. Cohn's theorem is useful for s

Abel–Ruffini theorem

In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitra

Multi-homogeneous Bézout theorem

In algebra and algebraic geometry, the multi-homogeneous Bézout theorem is a generalization to multi-homogeneous polynomials of Bézout's theorem, which counts the number of isolated common zeros of a

Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)

Equioscillation theorem

In mathematics, the equioscillation theorem concerns the approximation of continuous functions using polynomials when the merit function is the maximum difference (uniform norm). Its discovery is attr

Routh–Hurwitz theorem

In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz stable polynom

Bernstein's theorem (polynomials)

Bernstein's theorem is an inequality relating the maximum modulus of a complex polynomial function on the unit disk with the maximum modulus of its derivative on the unit disk. It was proven by Sergei

Lagrange's theorem (number theory)

In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime. More precisely, i

Schwartz–Zippel lemma

In mathematics, the Schwartz–Zippel lemma (also called the DeMillo-Lipton-Schwartz–Zippel lemma) is a tool commonly used in probabilistic polynomial identity testing, i.e. in the problem of determinin

Gauss's lemma (polynomials)

In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a statement about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a uniq

Descartes' rule of signs

In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for getting information on the number of positive real roots of a polynomial. It as

Kharitonov's theorem

Kharitonov's theorem is a result used in control theory to assess the stability of a dynamical system when the physical parameters of the system are not known precisely. When the coefficients of the c

Sturm's theorem

In mathematics, the Sturm sequence of a univariate polynomial p is a sequence of polynomials associated with p and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem ex

Gauss–Lucas theorem

In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial P and the roots of its derivative P′. The set of roots of a real or c

Mason–Stothers theorem

The Mason–Stothers theorem, or simply Mason's theorem, is a mathematical theorem about polynomials, analogous to the abc conjecture for integers. It is named after Walter Wilson Stothers, who publishe

Multinomial theorem

In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multino

Factor theorem

In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem. The factor theorem states that a polynomial has a facto

Rational root theorem

In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation with integer

Marden's theorem

In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with comp

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