faktorisiere

Faktorisiere is the German verb meaning to factorize, the mathematical process of expressing an object as a product of factors. In mathematics, factorization is used to rewrite expressions in a form that reveals their structure and simplifies further work such as solving equations, simplifying fractions, or integrating functions.

There are two primary contexts for factorization. Integer factorization decomposes a positive integer into a product of prime numbers. The fundamental theorem of arithmetic states that every integer greater than 1 has a unique prime factorization up to the order of the factors. Algorithms for integer factorization range from simple trial division to more advanced methods such as the Sieve of Eratosthenes, Pollard’s rho, Pollard p−1, and elliptic-curve factorization. Integer factorization is a central topic in number theory and underpins the security of many cryptographic systems, including RSA.

Polynomial factorization aims to write a polynomial as a product of irreducible polynomials over a chosen field, such as the rationals, reals, or finite fields. Over the rationals, many polynomials can be factored into rational factors; over finite fields, specialized algorithms like Berlekamp’s algorithm or Cantor–Zassenhaus are used, sometimes followed by lifting the factorization to other coefficient domains. Factorization is a key tool in solving equations, simplifying expressions, and analyzing algebraic structures.

In German-language education, factorization is taught as a core algebra skill, covering prime factorization of integers and the factorization of polynomials such as x^2 − 5x + 6 = (x − 2)(x − 3).