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Math Help - Find the missing factor

  1. #1
    r45
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    Find the missing factor

    Hi there,

    I am trying to find the missing factor here:

    x^n - k^n = (x - k)(?)

    I have tried various things. First I tried to do it by construction, giving:

    x^n - k^n = (x - k)(x^{n-1} + k^{n-1} + b)

    where b is such that:

    xb - kb = kx^{n-1} - xk^{n-1}

    But I can't find such a b.

    Then I tried to do it by long division of polynomials but because the exponent is n and not a given value, this got very messy and I couldn't get it work.

    Can someone help me out?

    Many thanks
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  2. #2
    r45
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    In fact, I could say from:

    xb - kb = kx^{n-1} - xk^{n-1}<br />

    [COLOR=black]That:

    b(x-k) = kx^{n-1} - xk^{n-1}

    b = ( kx^{n-1} - xk^{n-1} ) divided by (x-k) (i don't know how to do division using this formatting!)

    But inserting this into the identity gives a fairly trivial result and I don't think that's what the question is after...?
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  3. #3
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    x^n - k^n = (x - k)(x^{n-1} + kx^{n-2} + k^2x^{n-3} + ... + k^{n-3}x^2 + k^{n-2}x + k^{n-1})

    To verify this result, multiply it out and you will find that each term has a term that cancels with it except for the x^n and constant terms.
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  4. #4
    r45
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    Ahhh thankyou. I didn't think it would be something finite and clean, this makes much more sense. Many thanks and best wishes.
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