faktorisere

Faktorisere is the process of expressing a mathematical object as a product of factors. In number theory, factoring a positive integer means writing it as a product of primes, known as prime factorization. In algebra, factoring a polynomial means expressing it as a product of irreducible polynomials over a chosen coefficient field. For objects in rings such as the integers or polynomials over a field, factorization is typically unique up to multiplication by units and the order of the factors, reflecting the fundamental theorem of arithmetic for integers and the concept of a unique factorization domain for polynomials.

When factoring integers, common goals include finding the prime components and simplifying arithmetic operations, proofs, or cryptographic analyses. Simple methods include trial division, which tests divisibility by consecutive primes, and more advanced algorithms such as Pollard's rho, the quadratic sieve, or the elliptic curve method, which are used for larger numbers. The difficulty of prime factorization underpins the security of RSA encryption.

Factoring polynomials involves expressing a given polynomial as a product of lower-degree polynomials with coefficients in a specified field, often the rational numbers or real numbers. Techniques include the factor theorem and synthetic division to find roots, Gauss's lemma and content considerations, and specialized algorithms for factoring over finite fields (such as Berlekamp’s or Cantor–Zassenhaus). For example, x^2 − 5x + 6 factors as (x − 2)(x − 3).

In theory, factorization underpins the structure of numbers and polynomials, with practical implications in cryptography, algebraic solving, and symbolic computation.