In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor...

polynomials (see Polynomial greatest common divisor) and other commutative rings (see § In commutative rings below). The greatest common divisor (GCD) of integers...

and greatest common divisors of such polynomials. Gauss's lemma asserts that the product of two primitive polynomials is primitive. (A polynomial with...

their greatest common divisor. Extended Euclidean algorithm also refers to a very similar algorithm for computing the polynomial greatest common divisor and...

the Euclidean algorithm for polynomials that computes a polynomial greatest common divisor of two polynomials. Here, "greatest" means "having a maximal degree"...

generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. Gröbner bases...

algorithm for computing the polynomial greatest common divisor is a special case of Buchberger's algorithm restricted to polynomials of a single variable. Gaussian...

who proved it for polynomials, is the following theorem: Bézout's identity — Let a and b be integers with greatest common divisor d. Then there exist...

nonzero polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its...

arithmetic. It is used by the most efficient implementations of polynomial greatest common divisor, exact linear algebra and Gröbner basis algorithms over the...

performing Euclidean polynomial division Ruffini's rule Euclidean domain Gröbner basis Greatest common divisor of two polynomials Archived at Ghostarchive...

performed within the same time bounds. The cost of a polynomial greatest common divisor between two polynomials of degree at most n can be taken as O(n2) operations...

needs of the simplifier. For example, the computation of polynomial greatest common divisors is systematically used for the simplification of expressions...

those of f, and the greatest common divisor of two polynomials is independent of the ambient field, so the greatest common divisor of f and f′ has coefficients...

account. Other applications of multi-modular arithmetic include polynomial greatest common divisor, Gröbner basis computation and cryptography. A residue numeral...

1} is a greatest common divisor of the polynomial and its derivative. A square-free decomposition or square-free factorization of a polynomial is a factorization...

polynomials are well-behaved, and thus the formula given by the theorem for the number of their intersections is valid, if their polynomial greatest common...

or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both...

primitive polynomial. A primitive polynomial is a polynomial over a unique factorization domain, such that 1 is a greatest common divisor of its coefficients...

approximant is via the extended Euclidean algorithm for the polynomial greatest common divisor. The relation R ( x ) = P ( x ) / Q ( x ) = T m + n ( x )...

P} and Q {\displaystyle \textstyle Q} have a non-constant polynomial greatest common divisor R {\displaystyle \textstyle R} , then setting P = P 1 R {\displaystyle...

generated by the greatest common divisor of these minors. As one is working with polynomials with integer coefficients, this greatest common divisor is defined...

near John Day, Oregon, United States Greatest common divisor Binary GCD algorithm Polynomial greatest common divisor Lehmer's GCD algorithm Griffith College...

number of input numbers. The Euclidean algorithm for computing the greatest common divisor of two integers is one example. Given two integers a {\displaystyle...

of a rational number and a polynomial with integer coefficients, which is primitive (that is, the greatest common divisor of the coefficients is 1), and...

replace Euclidean division by pseudo-division for computing polynomial greatest common divisors. This amounts to replacing the remainder sequence of the...

divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors...

one half we get a factor of n by finding the greatest common divisor of n and x − y. The choice of polynomial can dramatically affect the time to complete...

quartic has a double root, it can be found by taking the polynomial greatest common divisor with its derivative. Then they can be divided out and the...

properties such as the existence of a Euclidean algorithm for computing greatest common divisors, Bézout's identity, the principal ideal property, Euclid's lemma...