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The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b). 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.
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain.
If one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form (x − r)Q(x) where Q(x) is a polynomial of degree n − 1. Q(x) is simply the quotient obtained from the division process; since r is known to be a root of P(x), it is known that the remainder must be zero.
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: =. That difference is algebraically factorable as (+) (); if neither factor equals one, it is a proper factorization of N.
In algebra, the elementary divisors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If R {\displaystyle R} is a PID and M {\displaystyle M} a finitely generated R {\displaystyle R} -module, then M is isomorphic to a finite direct sum of the form
When the denominator b(x) is monic and linear, that 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 a(c). [18] In this case, the quotient may be computed by Ruffini's rule, a special case of synthetic division. [20]
[6]: 202–207 If one is given a quadratic equation in the form x 2 + bx + c = 0, the sought factorization has the form (x + q)(x + s), and one has to find two numbers q and s that add up to b and whose product is c (this is sometimes called "Vieta's rule" [7] and is related to Vieta's formulas). As an example, x 2 + 5x + 6 factors as (x + 3)(x ...
Starting from two polynomials a and b, Euclid's algorithm consists of recursively replacing the pair (a, b) by (b, rem(a, b)) (where "rem(a, b)" denotes the remainder of the Euclidean division, computed by the algorithm of the preceding section), until b = 0. The GCD is the last non zero remainder.