quartic

Quartic refers to anything of degree four in mathematics. A quartic polynomial has degree four, generally written as ax^4 + bx^3 + cx^2 + dx + e with a ≠ 0. A quartic equation is an equation of the form ax^4 + bx^3 + cx^2 + dx + e = 0. The term also appears in geometry and analysis, such as quartic curves (algebraic curves of degree four) and quartic surfaces.

Form and properties: The leading coefficient a governs end behavior; as x approaches plus or minus infinity, f(x) behaves like a x^4, so it tends to positive infinity if a > 0. If the cubic and linear terms vanish (b = d = 0), the equation is biquadratic: ax^4 + cx^2 + e = 0, which reduces by setting y = x^2 to a quadratic in y.

Solving: General quartic equations are solvable by radicals. Ferrari's method reduces a depressed quartic (with no cubic term) to a resolvent cubic, whose solution yields a factorization of the quartic into two quadratics, from which the roots are obtained. The process is algorithmic but intricate; historically, Cardano developed methods for cubics, and Ferrari extended to quartics in the 16th century.

Applications and examples: Quartics arise in various problems, such as beam theory in engineering, optics for lens design, and physics. They are also important in algebraic geometry as simple examples of higher-degree plane curves. For example, the biquadratic x^4 - 5x^2 + 6 = 0 factors as (x^2-2)(x^2-3) = 0, yielding x = ±√2, ±√3.