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Math Help - Solving Cubic Equations with p/q (Finding possible rational roots)

  1. #1
    cYn
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    Solving Cubic Equations with p/q (Finding possible rational roots)

    So here is the equation

    f(x) = 3x^3 - x^2 + 6x - 2
    f(p/q) = ...

    After plugging in p/q and doing some basic algebra, we would get:

    q(-p^2 + 6pq - 2q^2) = -3p^3

    Now the book says the possible rational roots are 1/3, 2/3, 1, 2

    I'm stuck on how to arrive the possible rational roots.
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  2. #2
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    The rational roots theorem says that the possible rational roots of 3x^3 - x^2 + 6x - 2 = 0 are formed from a factor of 2 divided by a factor of 3, with both negative and positive roots being possible. The factors of 2 are 1 and 2, and the factors of 3 are 1 and 3. This is where you get the eight possibilities \pm \frac{1}{3}, \pm \frac{2}{3}, \pm 1, \pm 2.
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