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Then, take the product of all common factors. At this stage, we do not necessarily have a monic polynomial, so finally multiply this by a constant to make it a monic polynomial. This will be the GCD of the two polynomials as it includes all common divisors and is monic. Example one: Find the GCD of x 2 + 7x + 6 and x 2 − 5x − 6.
For example, 6 and 35 factor as 6 = 2 × 3 and 35 = 5 × 7, so they are not prime, but their prime factors are different, so 6 and 35 are coprime, with no common factors other than 1. A 24×60 rectangle is covered with ten 12×12 square tiles, where 12 is the GCD of 24 and 60.
If one uses the Euclidean algorithm and the elementary algorithms for multiplication and division, the computation of the greatest common divisor of two integers of at most n bits is O(n 2). This means that the computation of greatest common divisor has, up to a constant factor, the same complexity as the multiplication.
A third difference is that, in the polynomial case, the greatest common divisor is defined only up to the multiplication by a non zero constant. There are several ways to define unambiguously a greatest common divisor. In mathematics, it is common to require that the greatest common divisor be a monic polynomial.
The primitive part of a greatest common divisor of polynomials is the greatest common divisor (in R) of their primitive parts: ( (,)) = ( (), ()). The complete factorization of a polynomial over R is the product of the factorization (in R ) of the content and of the factorization (in the polynomial ring) of the primitive part.
Here, a greatest common divisor of a and b is an element d that divides both a and b, and such that every other common divisor of a and b divides d. All greatest common divisors of a and b are associated. Any UFD is integrally closed. In other words, if R is a UFD with quotient field K, and if an element k in K is a root of a monic polynomial ...
Here the greatest common divisor of 0 and 0 is taken to be 0.The integers x and y are called Bézout coefficients for (a, b); they are not unique.A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that | x | ≤ | b/d | and | y | ≤ | a/d |; equality occurs only if one of a and b is a multiple ...
Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials. Gauss's lemma asserts that the product of two primitive polynomials is primitive. (A polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients. [note 2])
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