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The elements 2 and 1 + √ −3 are two maximal common divisors (that is, any common divisor which is a multiple of 2 is associated to 2, the same holds for 1 + √ −3, but they are not associated, so there is no greatest common divisor of a and b.
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.
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial ...
x 2 − 5x − 6 = (12 x + 12) ( 1 / 12 x − 1 / 2 ) + 0 Since 12 x + 12 is the last nonzero remainder, it is a GCD of the original polynomials, and the monic GCD is x + 1 . In this example, it is not difficult to avoid introducing denominators by factoring out 12 before the second step.
If one of these values is 0, we have a linear factor. If the values are nonzero, we can list the possible factorizations for each. Now, 2 can only factor as 1×2, 2×1, (−1)×(−2), or (−2)×(−1). Therefore, if a second degree integer polynomial factor exists, it must take one of the values p(0) = 1, 2, −1, or −2. and likewise for p(1).
For example, the content of + may be either 2 or −2, since 2 is the greatest common divisor of −12, 30, and −20. If one chooses 2 as the content, the primitive part of this polynomial is If one chooses 2 as the content, the primitive part of this polynomial is
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm.
Euler's factorization method is a technique for factoring a number by writing it as a sum of two squares in two different ways. For example the number can be written as + or as + and Euler's method gives the factorization =.
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