faktörlüdür

Faktörlüdür is a Turkish mathematical adjective meaning “is factorable” or “can be factored.” The term is used to describe expressions, usually polynomials, that can be written as a product of two or more non-constant factors with coefficients in a specified ring or field. When an expression cannot be factored in this way, it is not faktörlüdür; in the context of polynomials over a given field, such an expression may be described as irreducible.

For a polynomial f(x) in a polynomial ring, faktörlüdür if there exist non-constant polynomials g(x) and h(x) such that f(x) = g(x) h(x). The factorization is not unique in general, but its irreducible factors are determined up to units and ordering.

Examples: The polynomial x^2 - 5x + 6 is faktörlüdür because it factors as (x-2)(x-3). Conversely, over the real numbers, x^2 + 1 is not faktörlüdür, since it cannot be written as a product of real-coefficient polynomials of smaller positive degree; over the complex numbers it becomes faktörlüdür as x^2 + 1 = (x - i)(x + i).

In teaching and algebraic theory, determining whether a given expression is faktörlüdür helps in solving equations by factoring, simplifying expressions, and understanding the structure of polynomial rings. Related concepts include factorization, irreducible polynomials, and the distinction between factoring over different coefficient domains.