The Cubic and Quartic Equations in Intermediate Algebra Courses

Josefino Villanueva

General College Division
Florida Memorial College
15800 Nw 42nd Ave
Miami, FL 33054
USA
Phone: (305) 626-3660
Fax: (305) 626-3664
jvillanu@fmc.edu

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ABSTRACT

The quadratic formula is well-known to students of intermediate algebra, but the solutions of the cubic or the quartic equations are not as familiar. Historically, however, the solutions to these equations have been known to renaissance mathematicians, and an analysis of their solutions shows that there is really not much more sophistication involved than radicals and complex numbers. Thus, these topics can be taught in our intermediate algebra courses to give our students a satisfying exercise in elementary analysis and impart an appreciation for the need for higher analysis, such as in abstract algebra. The drawback is that the closed form of their solution is rather lengthy and cumbersome that to deal with them thoroughly would take up at least two 50-minute class meetings in our already crowded syllabus in intermediate algebra. Other more practical methods would then be required.

This talk will first summarize the properties of the roots of the quadratic equation using the quadratic formula. This will then guide us in presenting the solution of the cubic, as was known in the 16th Century to Tartaglia, Cardano, Ferrari, and Descartes. A quick pass on the quartic will also be given. Other more practical methods of solution (such as Descartes' Rule of Signs, the Intermediate Value Theorem, the Rational Root Theorem, and other approximation techniques) will then be mentioned. Examples will be provided, as time permits. No original results are claimed, except the presentation of classical results.

Keyword(s): college algebra