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 _{n}x^{n}* +

*a*+ ... +

_{n}-1x^{n}-1*a*

_{2}

*x*

^{2}+

*a*

_{1}

*x*

^{1}+

*a*

_{}

*x*

^{}

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

*a _{n}x^{n}* +

*a*+ ... +

_{n}-1x^{n}-1*a*

_{2}

*x*

^{2}+

*a*

_{1}

*x*+

*a*

_{}

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

*a _{n}x^{n}* +

*a*+ ... +

_{n}-1x^{n}-1*a*

_{2}

*x*

^{2}+

*a*

_{1}

*x*+

*a*

_{}= 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.

*This was last updated in*May 2008

*Posted by:*Margaret Rouse

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