NotLKH: Assuming that your formula is correct, it still seems like a lot of work to find a root of a special case cubic.

It might be fun to develop such a formula, but it does not seem very practical. Newton-Raphson seems to be easy to program for any arbitrary cubic. Once you get one root of a cubic, it is fairly easy to determine the quadratic factor and find the other two roots.

I would not like to deal with anything higher than a quadratic without some programable device.

My HP calculator has a built in polynomial root finder that seems to work for order ten or more. I do not know what the limit is.

I developed a VB application which will find all roots (real & complex) of a polynomial. I have tried it for various polynomials up to order 21, and believe that it will do any order for which I am willing to enter the coefficients. I am not sure that I can accurately enter more than 30 or so coefficients.

BTW: As roots are found, my application uses synthetic division to reduce the order of the polynomial.

While all but the first one or two roots are found using reduced polynomials, Newton-Raphson versus the original polynomial is used to refine the precision of roots. I have discovered that roundoff errors in the coefficients of the reduced polynomials can result in significant loss of precision in the last 2-6 roots found for a high order polynomial.