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

In summary, the algorithm for division of polynomials can be typed in latex using the symbol \frac{...}{...}, as long as the coefficients of the divisor are properly aligned.
  • #1
paulmdrdo1
385
0
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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  • #2
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

$\chi$ $\sigma$
 
  • #3
Re: division of polynomials

what if i have something like

$9m^3n-32m^3+42m^6-15n^2-n+6$
 
  • #4
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 [tex]x,y,z[/tex], I took the terms with [tex]x^2,\,xy,\,xz[/tex]
. . [tex]x^2 - 8xy + 2xz[/tex]

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

So we have: .[tex]\frac{x^2-8xy + 2x z + 16y^2 - 8yz + z^2}{x-4y-z}[/tex][tex]\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}[/tex]Therefore: ..[tex]\frac{x^2-8xy + 2x z + 16y^2 - 8yz + z^2}{x-4y-z} \;\;=\;\;x - 4y + 3x + \frac{4z^2}{x-4y-z}[/tex]
 
  • #5
soroban said:
Hello, paulmdrdo!


It's tricky to explain my procedure . . .

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

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

So we have: .[tex]\frac{x^2-8xy + 2x z + 16y^2 - 8yz + z^2}{x-4y-z}[/tex][tex]\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}[/tex]Therefore: ..[tex]\frac{x^2-8xy + 2x z + 16y^2 - 8yz + z^2}{x-4y-z} \;\;=\;\;x - 4y + 3x + \frac{4z^2}{x-4y-z}[/tex]
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,
\(\displaystyle |\pi\rangle\)
 

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

What is the division of polynomials?

The division of polynomials is a mathematical operation that involves dividing a polynomial (a mathematical expression with one or more terms) by another polynomial. This results in a quotient and a remainder.

What is the process for dividing polynomials?

The process for dividing polynomials involves long division, similar to the long division used for dividing numbers. The divisor (the polynomial being divided by) is written on the left and the dividend (the polynomial being divided) is written on the right. The first term of the divisor is divided into the first term of the dividend, and the resulting term is written above the dividend. This process is repeated until all terms have been divided and a remainder is obtained, if necessary.

What are the rules for dividing polynomials?

There are several rules that apply when dividing polynomials. These include:

  1. The degree (highest exponent) of the divisor must be less than or equal to the degree of the dividend.
  2. If the degree of the divisor is less than the degree of the dividend, the quotient will have a degree that is equal to the difference between the two.
  3. If a term in the dividend has a higher degree than the corresponding term in the divisor, a placeholder of 0 is used in the quotient.
  4. If the divisor is a binomial (a polynomial with two terms), the division can be simplified using the distributive property.

Why is dividing polynomials important?

Dividing polynomials is important in various fields of science, including physics, engineering, and economics. It allows us to solve complex problems and equations, and is essential in finding the roots of polynomial functions. Additionally, understanding division of polynomials is crucial in understanding more advanced mathematical concepts such as rational functions and partial fractions.

What are some real-life applications of dividing polynomials?

Some real-life applications of dividing polynomials include calculating the motion of objects in physics, optimizing resources in economics, and designing circuits in engineering. It is also used in data analysis to determine trends and make predictions. Additionally, dividing polynomials is used in everyday situations such as dividing a bill among friends or calculating the average speed of a trip.

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