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Math Help - Help, it's hard! :-S

  1. #1
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    Help, it's hard! :-S

    In the ring Q[x], consider the polynomials:

    f(x) = x^5 + 2x^3 + x^2 + x + 1 and g(x) = x^4 + x^3 + x -1

    a) Find the greatest common divisor d(x) of f(x) and g(x).
    b) Find h(x), k(x) element in Q[x] so that h(x)f(x) + k(x)g(x) = d(x)
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  2. #2
    Senior Member
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    Let:
    <br />
f(x) = x^5 + 2x^3 + x^2 + x + 1<br />
    <br />
g(x) = x^4 + x^3 + x -1<br />

    Divide f(x) by g(x) to find:
    <br />
f(x) = g(x)(x-1) + 3(x^2+1)<br />

    Verify the remainder is zero in these divisions:
    <br />
g(x) = (x^2+1)(x^2+x-1)<br />
    <br />
h(x) = (x^2+1)(x^3+x+1)<br />

    So
    <br />
d(x) = (x^2+1)<br />

    Reinsert into the original division:
    <br />
d(x) = \frac{1}{3}( f(x) - g(x)(x-1) )<br />

    So
    <br />
h(x) = \frac{1}{3}<br />
    <br />
k(x) = -\frac{1}{3}(x-1)<br />
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