factorize

Factorize refers to expressing an object as a product of factors. In arithmetic, it most often means breaking a positive integer into a product of primes, a process called prime factorization. In algebra, factorization expresses a polynomial as a product of polynomials of smaller degree, ideally irreducible over a specified coefficient field. In abstract algebra, factorization is studied in structures where every element can be written as a product of irreducibles, with unique factorization properties in certain domains.

Examples help illustrate the idea. Integer factorization: 60 = 2^2 × 3 × 5. A larger example is 255 = 3 × 5 × 17. Polynomial factorization: x^2 − 5x + 6 factors as (x − 2)(x − 3) over the integers. Over the real numbers, x^2 + 1 is irreducible, while over the complex numbers it factors as (x − i)(x + i). A quartic such as x^4 − 5x^2 + 6 factors as (x^2 − 2)(x^2 − 3).

Methods vary by context. For integers, common techniques include trial division, wheel sieving, and more advanced algorithms for large numbers, such as Pollard’s rho or the general number field sieve. For polynomials, techniques include factoring by grouping, using the quadratic formula or rational root theorem to find linear factors, and various algorithms (for example, Berlekamp’s and Cantor–Zassenhaus) used in computer algebra systems.

Significance and applications: Factorization is central to the Fundamental Theorem of Arithmetic, which states that every positive integer has a unique prime factorization. In algebra, factoring simplifies expressions, aids in solving equations, and helps understand polynomial structure. In cryptography, the difficulty of factoring large integers underpins the security of several systems, notably RSA.