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Math Help - impossible question for me!!!!!

  1. #1
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    impossible question for me!!!!!

    when (x^4 - 6.x^3 + 16.x^2 - 25. x + 10) is divided by( x^2 - 2.x + k)
    the remainder is = x+a
    find the values of k and a
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by jashansinghal View Post
    When f=x^4 - 6.x^3 + 16.x^2 - 25. x + 10 is divided by g= x^2 - 2.x + k the remainder is r = x+a find the values of k and a
    There are, I am sure, many subtle ways of solving this problem, but the most obvious one which springs to mind is a simple cheat:

    We know that f=gh+r where h is some unknown polynomial and f, g, and r are as above, because that's what division with a remainder means. So, simply substitute unknowns into the polynomial h and then multiply out and pair off the coefficients, and you'll get your result.
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  3. #3
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    Quote Originally Posted by jashansinghal View Post
    when (x^4 - 6.x^3 + 16.x^2 - 25. x + 10) is divided by( x^2 - 2.x + k)
    the remainder is = x+a
    find the values of k and a
    After using long division, you should find

    \frac{x^4 - 6x^3 + 16x^2 - 25x + 10}{x^2 - 2x + k} = x^2 - 4x + 8 - k + \frac{(2k - 9)x + k^2 - 8k + 10}{x^2 - 2x + k}.


    So the remainder is:

    x + a = (2k - 9)x + k^2 - 8k + 10.


    Equating like co-efficients of x gives

    2k - 9 = 1 and k^2 - 8k + 10 = a.


    Solving for k in the first equation gives:

    2k = 10

    k = 5.


    Substituting into the second equation gives:

    5^2 - 8\cdot 5 + 10 = a

    25 - 40 + 10 = a

    a = -5.



    So a = -5 and k = 5.
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