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Expressing gcd of two polynomials as a linear combination

  1. Nov 23, 2013 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. Nov 24, 2013 #2

    Student100

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    You have the gcd as 1? What does that tell you about what factors these polynomials have in common?
     
  4. Nov 24, 2013 #3
    Just 1 but im having trouble writing the gcd as a linear combination of the two given polynomials to equal 1.
     
  5. Nov 24, 2013 #4

    Student100

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    Yeah, so they're coprime.

    You have [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^2-1) + 1 [/tex]

    back subbing you'd get [tex]

    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

    [/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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