Investigating Cardano's irreducible case.
INVESTIGATING CARDANO'S IRREDUCIBLE CASE. ALEX EDWARDS, UNDERGRADUATE, UNIV. OF NORTH ALA., FLORENCE, AL 35632. MICHEAL BEAVER, UNDERGRADUATE, UNIV. OF NORTH ALA., FLORENCE, AL 35632. DR. JESSICA STOVALL, DEPT. OF MATHEMATICS, UNIV. OF NORTH ALA., FLORENCE, AL 35632
Solving cubic equations is a historically rich problem in mathematics. Unlike with quadratic equations, cubic equations do not have a "cubic formula." However, over the years many techniques have been presented that find the solutions of cubic equations. Our research investigates one of these techniques known as Cardano's Method. This method provides an algebraic technique for solving the general cubic equation. Since its inception, this technique has suffered a significant drawback. In some instances, the application of Cardano's Method results in what Cardano termed the "irreducible case." The irreducible case occurs when a complex number is needed in order to complete the process. We are investigating the relationship among the coefficients of the general cubic equation and the irreducible case. We have determined that these relationships fall into one of three categories: always reducible, always irreducible, or conditionally irreducible. Through our research, we have discovered which relationships fall into each of the aforementioned categories. We are formulating a general algorithm to easily determine whether or not a given cubic equation will produce Cardano's irreducible case.
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|Title Annotation:||Physics and Mathematics Paper Abstracts|
|Author:||Edwards, Alex; Beaver, Micheal; Stovall, Jessica|
|Publication:||Journal of the Alabama Academy of Science|
|Date:||Apr 1, 2015|
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