The problem

Last time, in the March/April 2006 issue, we finished up the discussion of solving quadratic equations. This time we will begin a discussion of solving cubic equations. The conventional version of this problem is to find the roots x of the equation ( ) 3 23 3f x Ax Bx Cx D 0 = + + + = But what I am really interested in is the solution to homogeneous cubics: ( ) 3 2 2 3, 3 3f x w Ax Bx w Cxw Dw 0 = + + + = (0.1) These roots will consist of [x,w] pairs, where any scalar multiple represents the same root. Before we can solve this it’s important to first understand what types of solutions can occur and how they relate to the values of the four coefficients [A,B,C,D]. That’s primarily what we will do this time. The actual solution process will occur next time. Types of solutions There are four things that can happen with cubic polynomial roots. These consist of the various permutations of multiple roots and imaginary roots. There can be: • a triple root • a double root and a single root