section_path: Algebra/Polynomial equations
section_name: Polynomial equations
paragraph_text: Polynomial equations
  
  In general, an algebraic equation or polynomial equation is an equation of the form
  
  where P and Q are polynomials with coefficients in some field (e.g., rational numbers, real numbers, complex numbers). An algebraic equation is univariate if it involves only one 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,
  
  is a univariate algebraic (polynomial) equation with integer coefficients and
  
  is a multivariate polynomial equation over the rational numbers.
  
  Some polynomial equations with rational coefficients have a solution that is an algebraic expression, with a finite number of operations involving just those coefficients (i.e., can be solved algebraically). This can be done for all such equations of 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 or 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).