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Complex roots of the cubic are associated with vector plane rotation

Published online by Cambridge University Press:  17 February 2025

R. W. D. Nickalls
R. W. D. Nickalls*
Affiliation:
10 Queens Parade, Cheltenham GL50 3BB e-mail: [email protected]
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Extract

The all-real-roots configurations of the univariant cubic equation allow an easy visualisation of the roots as the orthogonal projection onto the real axis of the ends of three vectors of equal magnitude, radiating symmetrically in the XY plane from the origin of the reduced form [1].

Type
Articles
Information
The Mathematical Gazette , Volume 109 , Issue 574 , March 2025 , pp. 59 - 68
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Copyright
© The Authors, 2025 Published by Cambridge University Press on behalf of The Mathematical Association

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References

Nickalls, R. W. D., A new approach to solving the cubic: Cardan’s solution revealed, Math. Gaz. 77 (November 1993) pp. 354–359.CrossRefGoogle Scholar
Nickalls, R. W. D., A new bound for polynomials when all the roots are real, Math. Gaz. 95 (November 2011) pp. 520−526.CrossRefGoogle Scholar
Nickalls, R. W. D., The complementary cubic, Math. Gaz. 104 (March 2020) pp. 155−158.CrossRefGoogle Scholar
Holmes, G. C., The use of hyperbolic cosines in solving cubic polynomials, Math. Gaz. 86 (November 2002) pp. 473–477.CrossRefGoogle Scholar
Nickalls, R. W. D., On Note 92.35, Math. Gaz. 93 (March 2009) pp. 154–156.Google Scholar