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Thread: GCD of Polynomials

  1. #1
    Junior Member
    Feb 2010

    GCD of Polynomials

    I'm getting stuck on this problem.
    Find the GCD d(x) of the following polynomials f(x), g(x) in F[x], and express d(x)=s(x)f(x)+t(x)g(x) for appropriate s(x), t(x) en F[x]

    In this case, f(x)= x^2 + 2x + 2 and g(x)= x^2 + 1 for both F= the rationals and F= the complex number set.

    I started the problem by performing long division to see that f(x) / g(x) =1 with a remainder of 2x+1. I then realized that GCD of f(x), g(x) = GCD of g(x), 2x+1. But after division, I only got this result before getting stuck:

    x^2 + 1 = (2x+1)((1/2)x - (1/4)) + (3/4)

    Is this right so far, and how to do I finish it up? And how do I work this problem differently for the complexes?

    Thanks in advance!
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  2. #2
    Member Haven's Avatar
    Jul 2009
    Are you familiar with the Extended Euclidean Algorithm? If you are It can be easily generalized to polynomials
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  3. #3
    Junior Member
    Feb 2010
    I'm not familiar with the term "Extended Euclidean Algorithm" but I was trying to apply the method we used for the Euclidean Algorithm when I got stuck and couldn't figure out what my next step would be.

    I'm also confused as to what I do differently when F[x] is the rationals vs. the complex numbers.
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