Expressing gcd of two polynomials as a linear combination

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Homework Help Overview

The problem involves finding the greatest common divisor (gcd) of two polynomials, specifically ##gcd(x^3+x^2-x, x^5+x^4+2x^2-x-1)##, and expressing it as a linear combination of the two polynomials.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of the gcd being 1, indicating that the polynomials are coprime. There are attempts to express the gcd as a linear combination, with some participants sharing their steps and back substitution efforts.

Discussion Status

The discussion is ongoing, with participants exploring the relationships between the polynomials and the process of expressing the gcd as a linear combination. Some guidance has been offered regarding the steps taken, but there is no explicit consensus on the final expression.

Contextual Notes

Participants are navigating the challenge of writing the gcd as a linear combination while adhering to the constraints of the problem. There is a recognition of the difficulty in back substituting to achieve the desired form.

bonfire09
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Homework Statement



Find the ##gcd(x^3+x^2-x, x^5+x^4+2x^2-x-1) ##and write it as a linear combination.

Homework Equations


The Attempt at a Solution



I know the ##gcd(x^3+x^2-x, x^5+x^4+2x^2-x-1)=1## What I have so far is ##1. x^5+x^4+2x^2-x-1=(x^3+x^2-x)(x^2+1)+(x^2-1)## ##2. x^3+x^2-x=(x^2-1)(x+1)+1 ##. I tried back substituting but it can't seem to work. The division is correct.
 
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You have the gcd as 1? What does that tell you about what factors these polynomials have in common?
 
Just 1 but I am having trouble writing the gcd as a linear combination of the two given polynomials to equal 1.
 
bonfire09 said:
Just 1 but I am having trouble writing the gcd as a linear combination of the two given polynomials to equal 1.

Yeah, so they're coprime.

You have [tex] <br /> x^5+x^4+2x^2-x-1 =(x^2 + 1)(x^3+x^2-x) +(x^2 -1)[/tex]
[tex] x^3 + x^2 - x = (x+1)(x^2-1) + 1[/tex]

back subbing you'd get [tex] <br /> x^3 + x^2 -x -(x+1)(x^2-1) = 1[/tex]
[tex] x^5 + x^4 + 2x^2 -x -1 -(x^2+1)(x^3+x^2-x) = x^2-1[/tex]
[tex] x^3+x^2-x -(x+1)[x^5+x^4+2x^2 -x -1 -(x^2+1)(x^3+x^2-x)] = 1<br /> [/tex]

Does this help? I can't think of a good way to point you in the right direction without giving the soultion.
 

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