Algebraic equation

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In mathematics, an algebraic equation or polynomial equation is an equation of the form ${\displaystyle P=0}$, where P is a polynomial with coefficients in some field, often the field of the rational numbers. For example, ${\displaystyle x^{5}-3x+1=0}$ is an algebraic equation with integer coefficients and

${\displaystyle y^{4}+{\frac {xy}{2}}-{\frac {x^{3}}{3}}+xy^{2}+y^{2}+{\frac {1}{7}}=0}$

is a multivariate polynomial equation over the rationals. For many authors, the term algebraic equation refers only to the univariate case, that is polynomial equations that involve only one variable. On the other hand, a polynomial equation may involve several variables (the multivariate case), in which case the term polynomial equation is usually preferred.

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 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).