Polynomial root-finding

Polynomial root-finding is the problem of determining the roots of a univariate polynomial, either as numerical approximations or as closed-form expressions. It is a long-standing problem that has been studied throughout the history of mathematics and that influenced the development of irrational and complex numbers and of structures such as fields, rings and groups. Closed-form formulas exist only when the degree of the polynomial is less than 5, because the Abel-Ruffini theorem rules out a general formula in radicals at higher degrees, while the fundamental theorem of algebra guarantees that every nonconstant polynomial has at least one root. For that reason root-finding algorithms mostly produce numerical solutions, and different methods aim at a single root, at all complex roots at once, or at roots inside a chosen region of the complex plane. Although it no longer plays a central role in mathematics generally, it remains important in computer algebra.