This page contains an interactive tool that computes arithmetic relations between two integers or polynomials. It computes the greatest common divisor, the least common multiple, the Bezout relation and it performs successive Euclidean divisions.
Type of Material:
Classroom demonstration; student exploration
It requires a Java-enabled browser.
Identify Major Learning Goals:
To help in understanding several important concepts in algebra by performing step-by-step operations.
Target Student Population:
Abstract Algebra students.
Prerequisite Knowledge or Skills:
Basic understanding of algebra of polynomials.
Evaluation and Observation
Starting with beginning algebra courses, a great deal of time is devoted to developing skills in various manipulations with polynomials. This applet helps in understanding such important concepts of polynomial algebra as divisibility, greatest common divisor, least common multiple as well as the extended Euclidean algorithm and Bezout?s identity. (Bezout's identity states that if a and b are integers with greatest common divisor d, then there exist integers x and y such that ax + by = d. Bezout's identity works not only in the ring of integers, but also in any other principal ideal domain).
The applet works with either integers or polynomials which helps the user to draw parallels between these well known rings and generalize the result. Both input and output of the applet are very flexible and allow entering up to nine different polynomials (integers) while varying the desired output.
A warning that not all polynomials will be factored completely or that all factorizations are over the integers would help. For polynomials of degree 5 or higher it is understood by the expert that it is impossible to find all zeroes, but not necessarily by every student. Moreover, for polynomials of degree two, three and four, only the rational zeroes are found in the factorization. Irrational zeroes are not found, which can cause problems if we want to go into rings over the reals. Similarly, complex zeroes are not found. This is a concern for students who are not in an abstract algebra class, but who might be exposed to this applet at an earlier level. It gives the false impression that factoring only works with rational zeroes.
Potential Effectiveness as a Teaching Tool
Instructors can use the applet to illustrate at least four different concepts: gcd and lcm, factorization of polynomials, Euclidean algorithm and Bezout relation as part of an in-class presentation. The output of the module gives all the successive steps of the Euclidean algorithm and thus makes an excellent tool for a self-guided activity or homework preparation.
The site provides a crosslink to another similar tool by the same author that focuses on factorization of polynomials over the integers only.
Each concept discussed in this applet has a glossary link that takes the user directly to the search results in the on-line mathematical encyclopedia Wikipedia.
Two of the glossary links (gcd lcm and Bezout numbers) result in an empty search. The keyword "Bezout numbers" should be changed to "Bezout identity" in order to produce results.
Ease of Use for Both Students and Faculty
The applet is very easy to use. The average user can begin using the applet immediately. It contains both syntax references and examples.
One of the reviwers had problems loading framed version of example/help page. More explanations and working links to the encyclopedia are desirable.