1. Ferraris Method (square-completion of the even powered part of the
polynomial and then using a consistency condition to get a new cubic in an
2. Descartes Method (product of conjugate quadratics)
3. Lagrange's Method (using resolvent cubic)
4. I doubt there is one more method.
1. cardon's method:(original work by Scipione del Ferro (1465-1526), passed by Tartaglia to cardon).
2. Vieta's Geometric approach by seperating equations for real part and imaginary.
For quintics and above, I thought of using the singular perturbation technique to add extra terms that make the a polynomial to shrink to lower orders.
Over internet , I found some analysis for linear equations using singular quadratics .
I see some scope in this area , as Galois ruled out the Radical ways for Quintics , but still we can think of Non-Radical ways ,like this one. and one more important idea is to use transcendentals which don't comply with these finite polynomials.