derdegraads

Derdegraads refers to objects of the third degree, most often in mathematics to describe polynomials of degree three. A cubic polynomial has the general form f(x) = a x^3 + b x^2 + c x + d, where a ≠ 0 and the coefficients may be real or complex. The graph of a real cubic is a cubic curve, continuous and unbounded in both directions.

A cubic equation f(x) = 0 has at least one real root. Depending on the discriminant, it can have one real root plus two nonreal complex roots, or three distinct real roots (possibly with multiplicities). By removing the quadratic term with x = t − b/(3a), one obtains a depressed cubic t^3 + p t + q = 0, where p = (3 a c − b^2)/(3 a^2) and q = (2 b^3 − 9 a b c + 27 a^2 d)/(27 a^3). The discriminant Δ = −(4 p^3 + 27 q^2) governs the root pattern: Δ > 0 yields three real and distinct roots; Δ = 0 indicates multiple roots; Δ < 0 yields exactly one real root.

Cardano’s formula gives explicit expressions for the roots in terms of p and q, though the expressions involve cube roots of complex numbers in general. In practice, numerical methods such as Newton’s method are commonly used.

Applications: Derdegraads appear in algebra, geometry and numerical interpolation (cubic interpolation), physics and economics. They model curves with one or more bends and play a central role in polynomial factorization via the Rational Root Theorem and synthetic division.