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

AI Thread Summary
The discussion focuses on the proper arrangement of terms in polynomial division involving multiple variables, specifically how to organize them in descending powers. Participants express confusion about handling polynomials with different variable combinations, such as the given example of dividing a polynomial with three variables by a linear polynomial. A suggested method involves grouping terms based on their variable combinations to facilitate the division process. Additionally, there is mention of using LaTeX for formatting the division algorithm, with one participant recommending a video that clarifies polynomial division techniques. The conversation highlights the challenges and strategies in mastering polynomial division with multiple variables.
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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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