Analysis of the roots of some Cardano cubes

J. Leyendekkers and A. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 6, 2000, Number 4, Pages 113—117
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Authors and affiliations

J. Leyendekkers
The University of Sydney, 2006, Australia

A. Shannon
Warrane College, The University of New South Wales, 1465, &KvB Institute of Technology, North Sydney, 2060, Australia

Abstract

The Cardano cubic, y^3 - 6pqy - 3pq(p + q), p, q \in Z_{+}, has one real zero and a complex conjugate pair. The real zero is given by 2(2pq + e)^{1 \over 2} or (E + 2)(2pq)^{1 \over 2}, in which e, E are important parameters that feature in the roots of all Cardano cubics. They are functions of the coefficients of the complex conjugate pairs. We find that

    \[e = \frac{2q^2R \tan^2 \theta}{3 - \tan^2 \theta}\]

    \[E = 2 \Big \{ \Big ( \frac{3}{3 - \tan^2 \theta} \Big )^{1 \over 2} - 1 \Big \}\]

with R = \frac{p}{q} = h(\theta) and 11° < \theta < 60° for real zero. Furthermore, for E integer, the range of \theta is reduced to 52° < \theta < 60°, where the functional surfaces suggest the reason the integer E would only be compatible with an irrational value of R. This is verified algebraically.

AMS Classification

  • 11C08
  • 11D41

References

  1. Griffiths, H.B. and Hirst, A.E. 1994. Cubic Equations, or Where Did the Examination Question Come From? American Mathematical Monthly, 101.2: 151-161.
  2. Leyendekkers, J.V. and Shannon, A.G. The Cardano Family of Equations. Notes on Number Theory and Discrete Mathematics, submitted.
  3. Turnbull, H.W. 1957. Theory of Equations. Fifth Edition. Edinburgh: Oliver and Boyd.

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Cite this paper

Leyendekkers, J. & Shannon, A. (2000). Analysis of the roots of some Cardano cubes. Notes on Number Theory and Discrete Mathematics, 6(4), 113-117.

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