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How to find an unknown in a cubic equation iF you are given a factor?

  1. Mar 31, 2012 #1
    An example is x^3 + x^2 + ax -72
    Factor is x+3
    F(-3) doesnt equal zero and im out of other ideas. Help? :/
     
  2. jcsd
  3. Mar 31, 2012 #2
    If x+3 is a factor then F(-3) = 0. Why do you think F(-3) is not 0? What would make it 0?
     
  4. Mar 31, 2012 #3
    thats the problem, think without any exemple.
     
  5. Apr 1, 2012 #4

    SammyS

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    Hello Beurre. Welcome to PF !

    What is F(-3) ?
     
  6. Apr 1, 2012 #5
    Have you learned the relationship between roots and coefficients?
    [tex]S_n = x_1x_2...x_n = (-1)^n \dfrac{a_0}{a_n}[/tex]
    That is, the sum of the roots taken n at a time (in all possible combinations) equals the constant term divided by the nth coefficient multiplied by negative one raised to the nth power. I encourage you to research why this is true, so you don't blindly use the theorem. Regardless, let
    [tex]P(x) \textrm{ have roots } x_1, x_2, \textrm{ and } x_3[/tex]
    Then [tex] S_1 = x_1 + x_2 + x_3 ; S_2 = x_1x_2 + x_1x_3 + x_2x_3 ; S_3 = x_1x_2x_3[/tex]


    If not, and you are given at least one root and there is one coefficient missing, you can do synthetic division with x = -3 and deduce what that value must be.
    I can't really type synthetic division out here, but try doing it, because you know that the some value times a must equal 72.
     
    Last edited: Apr 1, 2012
  7. Apr 1, 2012 #6

    HallsofIvy

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    Beurre don't seem to have got back to us.

    Beurre, have you been able to do this? If not, since you assert that "F(-3) doesnt equal zero", what do you think it does equal?
     
  8. Apr 1, 2012 #7

    rcgldr

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    You could do long hand polynomial division of the cubic equation by the known factor in order to end up with a quadratic equation and a remainder that will be some linear function of a, which you can then solve for a, or as already suggested subsitute x = -3 into the equation and solve for a.
     
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