in Brioschi normal form: The iteration algorithm proceeds as follows: Set Compute the rational function where is a polynomial function given below, and is the derivative of with respect to Iterate on a random starting guess until it converges. Call the limit point and let . Compute where is a polynomial function given below. Do this for both and . Finally, compute for . These are two of the roots of the Brioschi quintic. The two polynomial functions and are as follows: This iteration method produces two roots of the quintic. The remaining three roots can be obtained by using synthetic division to divide the two roots out, producing a cubic equation. Due to the way the iteration is formulated, this method seems to always find two complex conjugate roots of the quintic even when all the quintic coefficients are real and the starting guess is real. This iteration method is derived from the symmetries of the icosahedron and is closely related to the method Felix Klein describes in his book. See also Theory of equations References Notes Other Sources External links Equations Polynomials Special functions