Solving polynomial equations

Every polynomial corresponds to a polynomial function, where f(x) is set equal to the polynomial, and to a polynomial equation, where the polynomial is set equal to zero. The solutions to the equation are called the roots of the polynomial and they are the zeroes of the function and the x-intercepts of its graph. If x = a is a root of a polynomial, then (x - a) is a factor of that polynomial.Some polynomials, such as f(x) = x2 + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every (non-constant) polynomial has at least one distinct root; this follows from the fundamental theorem of algebra.
There is a difference between approximating roots and finding exact roots. Formulas for the roots of polynomials up to a degree of 2 have been known since ancient times (see quadratic equation) and up to a degree of 4 since the 16th century (see Gerolamo Cardano, Niccolo Fontana Tartaglia). But formulas for degree 5 eluded researchers. In 1824, Niels Henrik Abel proved the striking result that there can be no general formula (involving only the arithmetical operations and radicals) for the roots of a polynomial of degree 5 or greater in terms of its coefficients (see Abel-Ruffini theorem). This result marked the start of Galois theory which engages in a detailed study of relationships among roots of polynomials.
Numerically solving a polynomial equation in one unknown is easily done on computer by the Durand-Kerner method or by some other root-finding algorithm. The reduction of equations in several unknowns to equations each in one unknown is discussed in the article on the Buchberger's algorithm. The special case where all the polynomials are of degree one is called a system of linear equations, for which a range of different solution methods exist, including the classical gaussian elimination
It has been shown by Richard Birkeland and Karl Meyr that the roots of any polynomial may be expressed in terms of multivariate hypergeometric functions. Ferdinand von Lindemann and Hiroshi Umemura showed that the roots may also be expressed in terms of Siegel modular functions, generalizations of the theta functions that appear in the theory of elliptic functions. These characterizations of the roots of arbitrary polynomials are generalizations of the methods previously discovered to solve the quintic equation.