Editing Quintic function (section)

==Finding roots of a quintic equation==

Finding the [[Zero of a function|roots]] (zeros) of a given polynomial has been a prominent mathematical problem.

Solving [[Linear equation|linear]], [[Quadratic equation|quadratic]], [[Cubic equation|cubic]] and [[quartic equation]]s in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions. However, there is no [[algebraic expression]] (that is, in terms of radicals) for the solutions of general quintic equations over the rationals; this statement is known as the [[Abel–Ruffini theorem]], first asserted in 1799 and completely proven in 1824. This result also holds for equations of higher degree. An example of a quintic whose roots cannot be expressed in terms of radicals is {{math| ''x''{{sup|5}} − ''x'' + 1 {{=}} 0}}.

Numerical approximations of quintics roots can be computed with [[Polynomial root-finding algorithms|root-finding algorithms for polynomials]]. Although some quintics may be solved in terms of radicals, the solution is generally too complicated to be used in practice.