About the Polynomials in Two Variables Demonstrating the Characteristics of Polynomials by Using a Model

Abstract

This research paper presents a new model of polynomials in two variables. By making a computer program of this model we can find many characteristics of polynomials, for example, Pascal's triangle, remainder theorem and many unexpected characteristics of polynomials.

2. A Model of Polynomials in Two Variables

At first we think a progression {a_i,j} for the polynomial =83=B0a_i,j xⁱ y^j and arrange a {a_i,j} to the plane. For example, coefficients of polynomial 3x² y -5xy² +2xy +4 is arranged as follows.(fig.1) Axis of abscissa means degree of x and axis of ordinates means degree of y. In the case of symmetrical polynomial, we see numbers are arranged symmetrically like in fig.2 and symmetorical axis is diagonal line. (i.e.y=3Dx)

For example, 3x² y -5xy² +2xy +4 multiplied by x² y is 3x⁴ y² -5x³ y³ +2x³ y² +4x² y. We can explain this prosess by using this model. In this model 3x² y -5xy² +2xy +4 moves 2 steps toward the axis of abscissa and moves 1 step toward the axis of ordinates.(fig.3) In other words multiplying a monomial of which coefficient is one means a parallel translation of these numbers.

4. Multiplying Polynomials

For example, fig.4 means x² y+x² +x +1 multiplied by x² and fig.5 means x² y+x² +x +1 multiplied by xy. By using these models we can get an answer of x² y+x² +x +1 multiplied by (x²+xy) from fig.6. In this model multiplying a polynomials means the sum of some moved numbers.

5. CAI (Computer Assisted Instruction) Program

I wrote a computer program in BASIC. In this program we can input coefficients of polynomials in the form of fig.1-6 by using a mouse. Then the computer represents an answer of calculation of polynomials in this model form. By using this program we can see many unexpected characteristics of polynomials and these characteristics cannot be found on the paper calcuration. I taught my class about the polynomials by using this program. Students were interested in finding unexpected characteristics of polynomials.