polynomial

A polynomial is a mathematical expression consisting of a sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient. The simplest polynomials have one variable. A one-variable (univariate) polynomial of degree n has the following form:

a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x¹ + a₀x⁰

where the a's represent the coefficients and x represents the variable. Because x¹ = x and x⁰ = 1 for all complex numbers x, the above expression can be simplified to:

a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀

When an nth-degree univariate polynomial is equal to zero, the result is a univariate polynomial equation of degree n:

a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀ = 0

There may be several different values of x, called roots, that satisfy a univariate polynomial equation. In general, the higher the order of the equation (that is, the larger the value of n), the more roots there are.

A univariate polynomial equation of degree 1 (n = 1) constitutes a linear equation. When n = 2, it is a quadratic equation; when n = 3, it is a cubic equation; when n = 4, it is a quartic equation; when n = 5, it is a quintic equation. The larger the value of n, the more difficult it is to find all the roots of a univariate polynomial equation.

Some polynomials have two, three, or more variables. A two-variable polynomial is called bivariate; a three-variable polynomial is called trivariate.