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

In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then...

redundancy check uses the remainder of polynomial division to detect errors in transmitted messages. Polynomial remainder theorem Synthetic division, a more...

Remainder theorem may refer to: Polynomial remainder theorem Chinese remainder theorem This disambiguation page lists mathematics articles associated with...

{\displaystyle a} is a root of the polynomial). The theorem is a special case of the polynomial remainder theorem. The theorem results from basic properties...

(integer division). In algebra of polynomials, the remainder is the polynomial "left over" after dividing one polynomial by another. The modulo operation...

is, b(x) = x − c for some constant c, then the polynomial remainder theorem asserts that the remainder of the division of a(x) by b(x) is the evaluation...

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 expresses...

Integer-valued polynomial Algebraic equation Factor theorem Polynomial remainder theorem See also Theory of equations below. Polynomial ring Greatest common...

following theorem: Given two univariate polynomials a and b ≠ 0 defined over a field, there exist two polynomials q (the quotient) and r (the remainder) which...

fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with...

division theorem can be generalized to univariate polynomials over a field and to Euclidean domains. In the case of univariate polynomials, the main...

(R(x)) polynomial, the degree of which is one less than that of P(x). The final value obtained, s, is the remainder. The polynomial remainder theorem asserts...

calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree k {\textstyle...

q(x) is some quotient polynomial and r(x) is a remainder polynomial such that 0 ≤ deg r(x) < n. By the Cayley–Hamilton theorem, replacing x by the matrix...

consequence of the polynomial remainder theorem, the entries in the third row are the coefficients of the second-degree polynomial, the quotient of f...

theorem (abstract algebra) Joubert's theorem (algebra) Lagrange's theorem (number theory) Mason–Stothers theorem (polynomials) Polynomial remainder theorem...

= P mod m 1 {\displaystyle R_{1}=P{\bmod {m}}_{1}} using the Polynomial remainder theorem, which can be done in O ( n log ⁡ n ) {\displaystyle O(n\log...

especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally...

a system of simultaneous polynomial congruences, and may be solved by means of the Chinese remainder theorem for polynomials. Birkhoff interpolation is...

Chinese remainder theorem. Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided...

1922. Theorem 1 (Schwartz, Zippel). Let P ∈ R [ x 1 , x 2 , … , x n ] {\displaystyle P\in R[x_{1},x_{2},\ldots ,x_{n}]} be a non-zero polynomial of total...

In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers...

division is useful in the context of the polynomial remainder theorem for evaluating univariate polynomials. To summarize, the value of p ( x ) {\displaystyle...

Bézout's lemma), named after Étienne Bézout who proved it for polynomials, is the following theorem: Bézout's identity—Let a and b be integers with greatest...

{\displaystyle \sum _{i=0}^{m+n}v_{i}X^{i}=Q(X).} The auxiliary polynomial theorem states max 0 ≤ i ≤ m + n ( | u i | , | v i | ) ≤ 2 b 9 ( m + n ) ...

points, one obtains the simple mean value theorem. Let P {\displaystyle P} be the Lagrange interpolation polynomial for f at x0, ..., xn. Then it follows...

coefficients. The primitive part of such a polynomial is the quotient of the polynomial by its content. Thus a polynomial is the product of its primitive part...

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories...

factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic...