Solving Cubic Equations Using Direct Factoring in Complex Field

Abstract

Cardano’s Method is not easily understood by undergraduate students. In this research project, we developed a method that students can understand without advanced mathematics skills. The method we developed only need to use the skills of factoring polynomials in complex field and finding cubic roots of a complex number.
The procedures we developed are as following:
1) Write cubic equation in the form of A3 + B3 + C3 – 3ABC = 0, where A is a function of x, B and C are complex numbers.
2) Solve the quadratic equation Z2 – (B3 + C3)Z + B3 C3 = 0, which gives B and C.
3) Factor equation of (1) into (A + B + C)(A +Bw + Cw2)(A +Bw2+Cw) = 0, where w is a complex root
of 1.
4) Solve equations A+B+C = 0, A+Bω+Cω2 = 0, and A+Bω2+Cω = 0