Editing Equation (section)

===Polynomial equations===
{{main|Polynomial equation}}

[[File:Polynomialdeg2.svg|thumb|right|220px|The solutions –1 and 2 of the polynomial equation {{nowrap|1=''x''<sup>2</sup> – ''x'' + 2 = 0}} are the points where the [[graph of a function|graph]] of the [[quadratic function]] {{nowrap|1=''y'' = ''x''<sup>2</sup> – ''x'' + 2}} cuts the x-axis.]]

In general, an ''algebraic equation'' or [[polynomial equation]] is an equation of the form

:<math>P = 0</math>, or

:<math>P = Q</math>{{Efn|As such an equation can be rewritten {{math|1=''P'' – ''Q'' = 0}}, many authors do not consider this case explicitly.}}

where ''P'' and ''Q'' are [[polynomial]]s with coefficients in some [[field (mathematics)|field]] (e.g., [[Rational number|rational numbers]], [[Real number|real numbers]], [[Complex number|complex numbers]]). An algebraic equation is ''univariate'' if it involves only one [[variable (mathematics)|variable]]. On the other hand, a polynomial equation may involve several variables, in which case it is called ''multivariate'' (multiple variables, x, y, z, etc.).

For example,
:<math>x^5-3x+1=0</math>
is a univariate algebraic (polynomial) equation with integer coefficients and
:<math>y^4+\frac{xy}{2}=\frac{x^3}{3}-xy^2+y^2-\frac{1}{7}</math>
is a multivariate polynomial equation over the rational numbers.

Some polynomial equations with [[Rational number|rational coefficients]] have a solution that is an [[algebraic expression]], with a finite number of operations involving just those coefficients (i.e., can be [[Algebraic solution|solved algebraically]]). This can be done for all such equations of [[Degree of a polynomial|degree]] one, two, three, or four; but equations of degree five or more cannot always be solved in this way, as the [[Abel–Ruffini theorem]] demonstrates. 

A large amount of research has been devoted to compute efficiently accurate approximations of the [[real number|real]] or [[complex number|complex]] solutions of a univariate algebraic equation (see [[Root finding of polynomials]]) and of the common solutions of several multivariate polynomial equations (see [[System of polynomial equations]]).