How to Properly Arrange Dividend in Polynomial Division with Multiple Variables?

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Discussion Overview

The discussion revolves around the proper arrangement of terms in polynomial division involving multiple variables. Participants explore the challenges of organizing polynomials with varying numbers of variables and the application of division algorithms, particularly in the context of LaTeX formatting for mathematical expressions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about arranging terms in descending powers of variables when dealing with polynomials that have multiple variables.
  • Another participant provides an example polynomial and asks for guidance on arranging terms correctly.
  • Some participants discuss their procedures for organizing terms based on the variables present, specifically focusing on how to handle terms with different variable combinations.
  • There is mention of using a division algorithm similar to arithmetic division and a request for assistance in formatting this process using LaTeX.
  • A participant shares a specific polynomial example to illustrate their question about arrangement.
  • One participant describes their approach to polynomial division and shares a detailed breakdown of their steps, including the use of LaTeX for clarity.
  • Another participant recommends a video resource that helped them understand polynomial division better, indicating a shared struggle with the concept.

Areas of Agreement / Disagreement

Participants generally express similar challenges regarding the arrangement of terms in polynomial division, but no consensus is reached on a definitive method or solution. Multiple approaches and examples are presented without resolution of the underlying confusion.

Contextual Notes

Participants highlight the complexity of polynomial division with multiple variables, noting that the arrangement of terms can depend on the specific variables involved and their powers. There is an acknowledgment of the difficulty in understanding the division process, particularly for those new to the topic.

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i was trying to solve this problem when i got confused on how to arrange the terms in descending powers of the literal factors because some term contain two variables and the polynomial has 3 variables. how can i properly arrange the dividend here?

$\displaystyle \frac{2xz-8xy-8yz+z^2+x^2+16y^2}{x-4y-z}$

and also I'm solving this using the algorithm used in arithmetic division is there a way to type that using latex?
 
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Re: division of polynomials

paulmdrdo said:
i was trying to solve this problem when i got confused on how to arrange the terms in descending powers of the literal factors because some term contain two variables and the polynomial has 3 variables. how can i properly arrange the dividend here?

$\displaystyle \frac{2xz-8xy-8yz+z^2+x^2+16y^2}{x-4y-z}$

and also I'm solving this using the algorithm used in arithmetic division is there a way to type that using latex?

Take into account that...

$\displaystyle (a\ x + b\ y + c\ z)^{2} = a^{2}\ x^{2} + b^{2}\ y^{2} + c^{2}\ y^{2} + 2\ a\ b\ x\ y\ + 2\ a\ c\ x\ z\ + 2\ b\ c\ y\ z\ (1)$

Kind regards

$\chi$ $\sigma$
 
Re: division of polynomials

what if i have something like

$9m^3n-32m^3+42m^6-15n^2-n+6$
 
Hello, paulmdrdo!

I was trying to solve this problem when i got confused on
how to arrange the terms in descending powers of the literal factors
because some term contain two variables and the polynomial has 3 variables.
How can i properly arrange the dividend here?

$\displaystyle \frac{2xz-8xy-8yz+z^2+x^2+16y^2}{x-4y-z}$
It's tricky to explain my procedure . . .

Since the divisor has x,y,z, I took the terms with x^2,\,xy,\,xz
. . x^2 - 8xy + 2xz

Then I took the terms with y^2,\,yz
. . 16y^2 - 8yz + z^2

So we have: .\frac{x^2-8xy + 2x z + 16y^2 - 8yz + z^2}{x-4y-z}\begin{array}{cccccccccccccc} &&&&&& x &&& - & 4y & + & 3z \\ && --&--&--&--&--&--&--&--&--& --&-- \\ x-4y-z & | & x^2 &-& 8xy &+& 2xz &+&16y^2 &-& 8yz &+& z^2 \\ &&x^2 &-& 4xy &-& xz \\ && --&--&--&--&-- \\ &&& -& 4xy &+& 3xz &+& 16y^2 &-& 8yz &+& z^2 \\ &&& - & 4xy &+& & +& 16y^2 &+& 4yz \\ &&& --&--&--&--&--&--&--&--&--&-- \\ &&&&&& 3xz &&& - & 12yz &+& z^2 \\ &&&&&& 3xz &&& -& 12yz &-& 3z^2 \\ &&&&&& --&--&--&--&--&--&-- \\ &&&&&&&&&&&& 4z^2 \end{array}Therefore: ..\frac{x^2-8xy + 2x z + 16y^2 - 8yz + z^2}{x-4y-z} \;\;=\;\;x - 4y + 3x + \frac{4z^2}{x-4y-z}
 
soroban said:
Hello, paulmdrdo!


It's tricky to explain my procedure . . .

Since the divisor has x,y,z, I took the terms with x^2,\,xy,\,xz
. . x^2 - 8xy + 2xz

Then I took the terms with y^2,\,yz
. . 16y^2 - 8yz + z^2

So we have: .\frac{x^2-8xy + 2x z + 16y^2 - 8yz + z^2}{x-4y-z}\begin{array}{cccccccccccccc} &&&&&& x &&& - & 4y & + & 3z \\ && --&--&--&--&--&--&--&--&--& --&-- \\ x-4y-z & | & x^2 &-& 8xy &+& 2xz &+&16y^2 &-& 8yz &+& z^2 \\ &&x^2 &-& 4xy &-& xz \\ && --&--&--&--&-- \\ &&& -& 4xy &+& 3xz &+& 16y^2 &-& 8yz &+& z^2 \\ &&& - & 4xy &+& & +& 16y^2 &+& 4yz \\ &&& --&--&--&--&--&--&--&--&--&-- \\ &&&&&& 3xz &&& - & 12yz &+& z^2 \\ &&&&&& 3xz &&& -& 12yz &-& 3z^2 \\ &&&&&& --&--&--&--&--&--&-- \\ &&&&&&&&&&&& 4z^2 \end{array}Therefore: ..\frac{x^2-8xy + 2x z + 16y^2 - 8yz + z^2}{x-4y-z} \;\;=\;\;x - 4y + 3x + \frac{4z^2}{x-4y-z}
I had Also hard to understand long polynom division but after watching this video evrything made sense! So I Will recommend him to watch this and it should be easy to understand http://m.youtube.com/watch?v=l6_ghhd7kwQ
Ps. You Really got nice latex skill, if I would do that I would just screw up and give up:P

Regards,
$$|\pi\rangle$$
 

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