Cubics

Cubics refer to polynomials or equations of degree three. The standard polynomial is f(x) = ax^3 + bx^2 + cx + d with a ≠ 0. When set equal to zero, it becomes a cubic equation. Cubic functions, given by y = f(x), are defined for all real x and are continuous and differentiable. They tend to infinity as x grows large in the positive or negative direction, and they typically have up to two turning points.

A cubic can have one or three real roots (counting multiplicities). By reducing to a depressed cubic t^3 + pt + q, the roots are classified by the discriminant Δ = -(4 p^3 + 27 q^2). If Δ > 0 there are three distinct real roots; Δ = 0 yields multiple real roots; Δ < 0 yields one real root and a pair of complex conjugate roots.

Solving methods range from factoring when possible and using the Rational Root Theorem with synthetic division, to explicit formulas for the general case. Cardano’s formula provides closed-form expressions for the roots of a cubic, although the expressions are lengthy. In practice, numerical methods such as Newton-Raphson iteration are commonly used to approximate real roots.

Geometrically, a cubic function has an inflection point at x = -b/(3a) and a derivative f'(x) = 3ax^2 + 2bx + c, which determines the stationary points. Cubics also appear in broader contexts, including algebraic geometry, where cubic curves serve as fundamental objects and, in certain cases, relate to elliptic curves when non-singular.