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Math Help - Find All of X

  1. #1
    Junior Member Dragon's Avatar
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    Find All of X

    Find all of x for which

    x^3-x^2-x+1
    ----------------- = 0
    x^3-x^2+x-1
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Dragon View Post
    Find all of x for which

    x^3-x^2-x+1
    ----------------- = 0
    x^3-x^2+x-1
    So long as the denominator isn't 0 for a particular value of x this equation may as well read
    x^3 - x^2 - x + 1 = 0
    (Just multiply both sides by x^3 - x^2 + x - 1)

    Just for reference when does x^3 - x^2 + x - 1 = 0?
    Let's apply the Rational Root theorem:
    Given a polynomial equation p*x^n + a*x^{n-1} + ... + bx + q = 0, the rational roots (if any exist) take the form of x = (+/-){factor of q}/{factor of p}.

    In this case p = 1 and q = -1. So the rational roots of x^3 - x^2 + x - 1 = 0, if any exist, take the form of x = (+/-)1. We see immediately that x = 1 is a root. Thus x^3 - x^2 + x - 1 = (x - 1)(x^2 + 1) = 0 (by long division.) So the other two roots are x = (+/-)I, where I^2 = -1.

    Thus our problem is
    x^3 - x^2 - x + 1 = 0, x not equal to -1, (+/-)I.

    Again, apply the rational root theorem.

    Any roots of this equation will take the form of x = (+/-)1. We see that x = 1 is again a root. Thus
    x^3 - x^2 - x + 1 = (x - 1)(x^2 - 1) = 0 (by long division)

    The remaining two solutions are thus x = (+/-)1.

    So our solution set is x = {-1, 1}, BUT we can't have x = -1. Thus the only solution to
    x^3-x^2-x+1
    ----------------- = 0
    x^3-x^2+x-1

    is x = 1.

    -Dan
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