COMPUTATIONAL TECHNIQUE FOR TEACHING MATHEMATICS (CTTM) - VISUALIZING THE POLYNOMIAL´S RESULTANT

Oct 15, 2026

Uploaded by Rizky

Computational Technique for Teaching Mathematics (CTTM) applies Dynamic System GeoGebra (DSG) to support learning and teaching of core mathematical concepts in Brazil, then extends discussion to academic-level topics on polynomial resultants. Using software-assisted exploration in algebraic geometry, the text analyzes when two plane algebraic curves share common factors and highlights limitations of GeoGebra. Maple is introduced when additional conclusive results are required. A structured proposal for CTTM is presented, combining analytical, numerical, and geometrical perspectives.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages12

Reading Time30 min

AccessFree to download

1. Introduction

Intersection and elimination theory

2. The Resultant

Determinant definition and properties

Example with Maple and real solutions

COMPUTATIONAL TECHNIQUE FOR TEACHING MATHEMATICS (CTTM): VISUALIZING THE POLYNOMIAL´ S RESULTANT Francisco Regis Vieira Alves Abstract: We find several applications of the Dynamic System Geogebra – DSG related predominantly to the basic mathematical concepts at the context of the learning and teaching in Brasil. However, all these works were developed in the basic level of Mathematics. On the other hand, we discuss and explore, with DSG´s help, some applications of the polynomial’s resultant atthe academic level. In this context, we have the opportunity to identify certain limitations of this software. In fact, when we consider the Algebraic Geometry (AG), some analytical methods permit to describe when two plane algebraic curves f and g have a common factor. So, we will show some cases that the CAS Maple is necessary too for providing some conclusive dates. Finally, with this example, we wish to describe/systematize and structure a proposal of a Computation Technique for Teaching Mathematics (CTTM) in the academic level. Key words: Computational Technique, Visualization, Polynomial’s resultant, Geogebra, Mathematics Education. 1. Introduction In Brazil, we find several works that are within the scope of Mathematcs Education – MA. These works are related to the use of technology in MA. In a specific branch of study, we observe a strong interest about the visualization of mathematical concepts, specially, in the academic level. In our work, we have pointed out the use of two softwares for specific academic content instruction (Alves, 2014a; 2014c; 2014f) . Moreover, several limitations and the didactical space of exploration permit identify some implications for the teaching and learning (Alves, 2014a; 2014b; 2014d; 2014e) . In Algebraic Geometry (AG), by a standard way, we seek to find the intersection multiplicities to their common points, in such a way that the Bezout´s classical theorem holds. For example, given two curves indicated by the following equations f (X, Y ) 􀀠 0 and g (X, Y ) 􀀠 0 , what can we talk about the intersection and the common points? We assume that these curves have no common components. We know a general mathematical methods that involves the one variable selections, we take X , for example, and figurate like a coefficient. Explicitly, we write f and g polynomials in a variable Y , with coefficients in the ringK[ X ] . Finally, we try to conclude if the polynomials f (x, Y ) and g (x, Y ) ( or f (X, y) and g (X, y) ) have a common factor. Moreover, from this method, we desire to determine the projections, over the axe Ox, related to the points f g . Mathematically, we called it by elimination’s theory and, we have a powerful tool called the two polynomials´ resultant (Dieudonné, 1974; Fulton, 1969) . In the next section, we will indicate some formal elements that we need to establish, en virtue our discussion. For this, the actual technology has a prominent role en virtue to describe several implications to the teaching and learning at the academic level. 2. The Resultant We consider a commutative ring A , for example A 􀀠 K[ X ] and take the polynomialsf 􀀠 ad Yd 􀀎 ad 􀀐1Yd 􀀐1 􀀎 􀀎 a0 and g 􀀠 beYe 􀀎 be􀀐1Ye􀀐1 􀀎 􀀎 b0 , with condition d 􀁴1 and e 􀁴1 . We define the resultant of the f , g by the following determinant (see Vainsencher, 2009, p. 22; Tignol, 2002, p. 56): Rf ,g 􀀠 det A(d 􀀎e)􀁵 (d 􀀎e) 􀀠 | ad ad 􀀐1 ad 􀀐2 | ad 􀀐3 a0 0 | 0 0 0 | 0

0

0

a0
0

0

b0 | | --- | --- | --- | --- | | ad ad 􀀐1 ad 􀀐2 a0 0 0 0 0
ad ad 􀀐1 a0 0 0 0 0
0 ad ad 􀀐1 ad 􀀐2 ad 􀀐3
be be􀀐1 be􀀐2 b0 0 0 0 0
0 be be􀀐1 be􀀐2 b0 0 0 0 0
0 0 0 0 0 be be􀀐1 be􀀐2 be􀀐3 | | | | We show a matrix determinant of order (d 􀀎 e) 􀁵 (d 􀀎 e) , with ‘e’ lines determined by a ´s and ‘d’lines determined by the terms b ´s . There are some spaces and we write the digit zero. In this definition, we observe some limitations en virtue to explore a mathematical software.

FAQs

What role does DSG (Dynamic System GeoGebra) play in the proposed CTTM approach?

The document uses DSG primarily to visualize and support learning of foundational mathematical concepts, then extends its use toward academic applications involving polynomial resultants.

How does the text connect polynomial resultants to algebraic geometry teaching?

It explains that resultants, via elimination theory, help determine when two plane algebraic curves have common factors, supporting discussion of intersection behavior and shared components.

Why is CAS Maple needed in the example?

The paper notes that GeoGebra may not provide conclusive analytical data for certain academic-level computations, so Maple is used to obtain roots and confirm the real numerical solutions.

Oct 15, 2026

Uploaded by Rizky