seventhdegree

Seventhdegree is a term used in mathematics and related disciplines to denote a quantity, object, or equation of degree seven. In polynomials, a seventh-degree polynomial is a polynomial whose highest exponent is 7.

A seventh-degree polynomial has the general form P(x) = a7 x^7 + a6 x^6 + a5 x^5 + a4 x^4 + a3 x^3 + a2 x^2 + a1 x + a0, with a7 ≠ 0. Over the complex numbers, such a polynomial has exactly seven roots counting multiplicities (the Fundamental Theorem of Algebra). If the coefficients are real, complex roots occur in conjugate pairs, and the number of real roots is odd, with possible values 1, 3, 5, or 7.

In terms of solvability, a general seventh-degree equation is not solvable by radicals, according to Abel–Ruffini. However, specific seventh-degree polynomials may be solvable by radicals or by special methods, particularly those with symmetries or that factor into lower-degree components.

The adjective septic is used in geometry to denote seventh-degree curves, such as septic curves defined by seventh-degree equations. In music theory, the seventh degree of a diatonic scale is the leading tone in major and harmonic minor contexts, and the term is also used for the scale degree built on the seventh note.

The term derives from Latin septimus, meaning seventh. In general mathematical discourse, "seventh degree" emphasizes the degree rather than any particular instance, applicable to polynomials, algebraic equations, and related objects.

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