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Integral of rational functions

  1. Feb 18, 2004 #1
    Can someone pls help me solve this integral?

    integral of (6x^2-13x-43)dx/(x^3-1x^2-8x+12)

    it's supposed to be solved using partial fractions, but I am having trouble factoring the denom correctly so I can apply it...

    Thanks
     
  2. jcsd
  3. Feb 18, 2004 #2

    matt grime

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    2 appears to be a root of the denominator, but why did you specify the coeff of the x^2 term? it appears to be a typo because of it.
     
  4. Feb 18, 2004 #3

    Hurkyl

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    There's a very useful theorem about rational roots of an integer polynomial:


    Theorem: If [itex]y[/itex] is a rational number that is a root of the polynomial

    [tex]f(x) = a_0 + a_1 x + \ldots + a_n x^n \quad (a_n \neq 0)[/tex]

    Then [itex]y[/itex] can be written as [itex]p / q[/itex] for some integers [itex]p[/itex] and [itex]q[/itex] where [itex]p | a_0[/itex] and [itex]q | a_n[/itex].

    ([itex]a | b[/itex] means "a divides b")


    Using this theorem, if [itex]x^3 - x^2 - 8x + 12[/itex] has a rational root, then it can be written in the form [itex]p/q[/itex] where [itex]p \in \{1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12\}[/itex] and [itex]q \in \{1, -1\}[/itex]. Only 12 possibilities to try, so if one exists you can find it by exhaustion. :smile:

    When trying to factor large polynomials, this is usually a good place to start.
     
    Last edited: Feb 18, 2004
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