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Math Help - GCD of two polynomials

  1. #1
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    GCD of two polynomials

    My task is to find the gcd of nx^{n+1}-(n+1)x+1 and x^{n}-nx-1.

    Long division isn't giving me a clue here. My hunch is that the gcd is one, but I'm not sure I can prove that by showing they are irreducible (over what ring would I do that anyway? the question does not specify) or something like that.
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  2. #2
    Senior Member roninpro's Avatar
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    You can use the fact that \gcd(a,b) will divide ma+nb, where a,b,m,n are elements from your ring.

    Applying this, \gcd(nx^{n+1}-(n+1)x+1, x^{n}-nx-1) also divides [nx^{n+1}-(n+1)x+1]-nx[x^{n}-nx-1]=-(n+1)x+1+n^2x^2+nx=n^2x^2-x+1

    You can try to draw a conclusion from this.
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  3. #3
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    One more step?

    If n=2, you can repeat the subtraction:
    (n^2x^2-x+1) - n^2(x^n-nx-1) =
    (4x^2-x+1) - 4(x^2-2x-1) =
    7x+5
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