Harmadfokú

Harmadfokú is a Hungarian term meaning third-degree or cubic. It is used in mathematics to describe objects and relations of degree three, including polynomials, equations, and curves. The term contrasts with first- and second-degree concepts, such as elsőfokú (linear) and másodfokú (quadratic).

In algebra, a cubic polynomial has the general form p(x) = ax^3 + bx^2 + cx + d, where a ≠ 0. A cubic equation is written as p(x) = 0. Such equations have at least one real root, and up to three real roots (with the remaining roots potentially complex). Solutions can be obtained by factoring, substitution, or Cardano’s formula; when all roots are real, the casus irreducibilis may require complex expressions to represent the real roots. The discriminant Δ = 18abcd − 4b^3d + b^2c^2 − 4ac^3 − 27a^2d^2 helps determine the number of real roots.

In geometry and analysis, a cubic function is of the form y = ax^3 + bx^2 + cx + d. Its graph is a smooth curve with up to two turning points. Cubic curves can be singular or nonsingular; nonsingular cubic curves in the projective plane are central to the theory of elliptic curves, which have applications in number theory and cryptography.

In applications, cubic polynomials model non-linear phenomena and are used in interpolation as cubic splines. The term harmadfokú appears in Hungarian mathematical literature to designate third-degree objects, in contrast with other degrees such as elsőfokú or másodfokú.