Solving Polynomials

Notes for Teachers

The material contained on this page is useful for constructing lessons on solving polynomial equations, of quadratic, cubic or quartic order, by exact formulae. To use all of the material herein, students should know how to take roots of complex numbers, i.e., they should know De Moivre's theorem (although this is not necessary for most of the material). This material could be used in an honors Pre-calculus course, or a College Algebra/Trigonometry course.

History

Formulae exist which give the roots of quadratic, cubic and quartic polynomial equations (whose coefficients are real). The solution for quadratic polynomials, i.e., the Quadratic Formula, is well-known (apparently, since antiquity). The general formula for the roots of cubic polynomial equations is attributed to Nicolo Fontana Tartaglia (1500-1557) and Girolamo Cardano (1501-1576), two Italian mathematicians. The solution for the roots of a quartic polynomial is due to Lodovico Ferrari (1522-1565), also an Italian mathematician. Finally, it was proven, in 1824, that general formulae do not exist for the roots of quintic and higher order polynomial equations, see the Abel-Ruffini theorem.

The general formulae which give the roots to quadratic, cubic and quartic polynomial equations (with real coefficients) are listed in the "Roots of Polynomials" document.

Derivations

If you are interested, the derivation of the formulae for the roots of cubic polynomials is given in "The Cubic Formula" document. Also, the derivation of the formulae for the roots of quartic polynomials is given in the "The Quartic Formula" document. Finally, the "Cubic Formula" can be written in a different form, which is the topic of the Alternate Form of the Cubic Formula page, and the "Quartic Formula" can be written more explicitly, which is the topic of the Alternate Form of the Quartic Formula page.

Examples

Example problems, which implement the formulae for the roots of cubic and quartic equations, are presented in the "Examples" document.

You may download a program, which reliably solves polynomial equations of orders two through eight (inclusive), by going to this page.