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Higher-degree equations

Évariste Galois and Niels Henrik Abel showed independently that in general a polynomial of degree 5 or higher is not solvable using radicals. Some particular equations do have solutions, such as those associated with the cyclotomic polynomials of degrees 5 and 17.

Charles Hermite, on the other hand, showed that polynomials of degree 5 are solvable using elliptical functions.

Otherwise, one may find numerical approximations to the roots using root-finding algorithms, such as Newton's method.