higherdegree

Higherdegree refers to polynomials whose degree is greater than two. In algebra, the degree of a polynomial is the largest exponent of its variable with a nonzero coefficient. Polynomials of degree three or more are called higher-degree polynomials; the specific names include cubic (degree 3), quartic (4), quintic (5), and so on.

Formally, a polynomial P(x) = a_n x^n + ... + a_1 x + a_0 with a_n ≠ 0 has degree n. The coefficient a_n is the leading coefficient. For example, P(x) = 2x^3 − x^2 + 5 has degree 3 (cubic).

Basic properties: The degree of a product equals the sum of degrees; deg(PQ) = deg P + deg Q if nonzero. The degree of a sum satisfies deg(P+Q) ≤ max(deg P, deg Q); equality can fail if cancellation occurs. The degree of the zero polynomial is not defined in standard convention.

Solving and theory: The Fundamental Theorem of Algebra states that a degree n polynomial over the complex numbers has exactly n roots counted with multiplicity. In contrast to quadratics, general formulas for roots do not exist for degree five and above (Abel–Ruffini). Practically, higher-degree polynomials are solved by factoring, numerical methods like Newton-Raphson, or approximations.

Applications: Higher-degree polynomials model nonlinear relationships in physics, economics, and engineering; they are central to polynomial interpolation, curve fitting, and algebraic geometry. They also appear in optimization and computer algebra systems.