Overview

The degree of a polynomial is a basic numerical measure of its highest power in the indeterminate(s). For a univariate polynomial, written for example as 2x3 − 7x2 + 5x − 4, the degree is 3 because the largest exponent of x that appears with a nonzero coefficient is 3. The degree is commonly denoted by deg(p) or simply deg when the polynomial is clear from context.

Definition and notation

Formally, if p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0 with coefficients a_i in a ring or field and a_n ≠ 0, then the degree of p is n and the term a_n x^n is the leading term; a_n is the leading coefficient. Constants (nonzero) have degree 0. The zero polynomial (all coefficients zero) is a special case: many texts leave its degree undefined or assign −∞ to reflect algebraic rules compactly. Notationally one writes deg(p) = n.

Basic properties

Degree behaves predictably in many settings, but with important caveats. Over an integral domain (for example the integers, rationals, or any field) the following hold:

  • Sum: deg(p + q) ≤ max(deg p, deg q); equality may fail if leading terms cancel.
  • Product: deg(p·q) = deg p + deg q for nonzero polynomials.
  • Scalar multiplication: multiplying by a nonzero scalar does not change degree.

If coefficients come from a ring with zero divisors, product degrees can drop for reasons of cancellation among coefficients.

For polynomials in several variables, several notions of degree exist. The total degree of a monomial is the sum of the exponents (for x^2 y^3 the total degree is 5). The degree of a multivariable polynomial is the maximum total degree among its monomials. One also considers the degree in a particular variable (partial degree) and homogeneous polynomials, which are those whose nonzero monomials all share the same total degree. These distinctions are important in algebraic geometry and in classifying polynomial systems.

Examples and simple rules

Examples illustrate the ideas: the polynomial −6y^5 + 2y^3 − 25y − 58 has degree 5 because of the y^5 term; the polynomial 7 has degree 0. If p(x)=x^2+1 and q(x)=x^2−1 then p+q=2x^2 has degree 2, while p−q=2 has degree 0 because the x^2 terms cancel. The Fundamental Theorem of Algebra connects degree to roots: a nonzero complex polynomial of degree n has exactly n complex roots counted with multiplicity.

Uses, importance and caveats

Degree is used to classify polynomials, predict asymptotic behavior, bound the number of roots, and analyze algorithms in computer algebra. It guides factorization, interpolation, and differential equation solutions. Care is required when working over rings with zero divisors or when cancellation changes the apparent degree of a sum or product. Many algorithms assume coefficients lie in a field to preserve degree arithmetic.

Historical and mathematical context

The notion of degree grew naturally with the algebra of polynomials as mathematicians organized expressions by powers. It underlies fundamental results in algebra and analysis and plays a central role in modern fields such as algebraic geometry, where the degree of a polynomial equation helps describe the geometry of its solution set, and computational mathematics, where degree affects complexity and storage.

Common distinctions and final remarks

It is useful to distinguish between degree, leading coefficient, and order of a zero (multiplicity). A root of multiplicity m reduces the number of distinct linear factors but counts m times toward the total degree. Remember that the name of the variable is irrelevant: replacing x by y does not change degree. For further structured definitions and examples see the entry on polynomial and treatments of degree in several variables.