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Math Help - gcd and polynomials

  1. #1
    Junior Member
    Joined
    Nov 2009
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    gcd and polynomials

    Find the greatest common divisor
    d(x) of
    f(x) = x^5 2x^4 3x^3 15x^2 + 10x + 25
    and
    g(x) = x^4 6x^2 24^x 35

    -----------

    using the polynomial long division

    f(x) / g (x)

    i ended up with 3x^3 + 135 x^2 -513x -815

    couldn't factorise further !

    ---

    the step after that

    x^4 6x^2 24^x 35 / 3x^3 + 135 x^2 -513x -815

    and the answer was -13275x^2 -53578x+9150



    but I don't know what should I do after that to get the gcd ,

    and I'm not sure if my work is alright ..


    any help ?

    thnx

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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, zis!

    Find the greatest common divisor of: . \begin{array}{ccc}f(x) \;=\; x^5 - 2x^4 - 3x^3 - 15x^2 + 10x + 25 \\ g(x) \;=\;x^4 - 6x^2 - 24x - 35 \end{array}

    I found that the GCD is: . x^2 + 2x + 5


    Then I could factor them:

    . . f(x) \;=\;(x^2+2x+5)(x^3 - 4x^2 + 5)

    . . g(x) \;=\;(x^2+2x+5)(x^2-2x - 7)


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    In desperation, I tried the Euclidean Algorithm on the polynomials
    After a slight modification, the answer popped out!

    Can anyone verify that the algorithm does apply to polynomials?

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