Polynomial equations

In mathematics, an algebraic equation or polynomial equation is an equation of the form

where P and Q are polynomials with coefficients in some field, often the field of the rational numbers. For most authors, an algebraic equation is univariate, which means that it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called multivariate and the term polynomial equation is usually preferred to algebraic equation.

For example,

is an algebraic equation with integer coefficients and

is a multivariate polynomial equation over the rationals.

Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations, not for all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).

The study of algebraic equations is probably as old as mathematics: the Babylonian mathematicians, as early as 2000 BC could solve some kinds of quadratic equations (displayed on Old Babylonian clay tablets).

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