Eric.ed.gov – Cubic Polynomials with Real or Complex Coefficients: The Full Picture

The cubic polynomial with real coefficients has a rich and interesting history primarily associated with the endeavours of great mathematicians like del Ferro, Tartaglia, Cardano or Vieta who sought a solution for the roots (Katz, 1998; see Chapter 12.3: The Solution of the Cubic Equation). Suffice it to say that since the times of renaissance mathematics in Italy various techniques have been developed which yield the three roots of a general cubic equation. Students studying a mathematics specialism as a part of the Victorian VCE (Mathematical Methods, 2010), HSC in NSW (Mathematics Extension in NSW, 1997) or Queensland QCE (Mathematics C, 2009) are expected to be able to recognise, plot, factorise, and even solve cubic polynomials as described by ACARA (ACARA, n.d., Unit 1, Topic 1: Functions and Graphs). The latter requirement even goes so far as to encourage students to “solve cubic equations using technology”. Since the application of such technology, e.g., MathCAD (2007), is fundamental to the contents of this paper, it is hoped that the work presented here will prove interesting, relevant, and stimulating to students and teachers alike. This article aims to allow all the roots of a cubic polynomial to be visualised, their arrangement better understood, and an explanation provided for the complex conjugate root pairing.