Quartics

Quartics are polynomials of degree four in one variable. The general form is ax^4 + bx^3 + cx^2 + dx + e, with a ≠ 0. When b and d are zero the quartic is biquadratic, easily reduced by substituting y = x^2. Over the complex numbers, a quartic has four roots counting multiplicities, by the Fundamental Theorem of Algebra.

Solving: The quartic equation can be solved in radicals. The standard approach, Ferrari's method, introduces an auxiliary variable to reduce the quartic to a product of two quadratics, after solving a resolvent cubic. This yields explicit but lengthy formulas for the four roots (real or complex). In practice one often uses numerical methods or factoring when possible.

Real roots and graph: End behavior depends on the leading coefficient a. If a > 0, the graph tends to +∞ on both ends; if a < 0, to −∞. A quartic can have up to three turning points and up to four real roots. The number of real roots is determined by how the graph intersects the x-axis and can be influenced by the discriminant in special cases.

Applications and related topics: Quartics appear in algebraic geometry, physics, engineering, and computer graphics, including problems of area, optimization, and curve intersections. They are connected to biquadratics, resolvent cubics, and various factoring strategies in polynomial algebra.