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Homework Help: Find the equation of cubic polynomial

  1. May 20, 2012 #1
    1. The problem statement, all variables and given/known data
    If α, β and γ are roots of cubic polynomial and:
    αβγ = 6

    Find the equation of cubic polynomial

    2. Relevant equations

    3. The attempt at a solution
    The equation is in the form:
    x3 - (α+β+γ)x2 + (αβ + αγ + βγ) x - αβγ = 0

    But I don't know how to find α+β+γ and αβ + αγ + βγ. Thanks
  2. jcsd
  3. May 20, 2012 #2


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    oh come on songoku! …

    (x-α)(x-β)(x-γ) ? :wink:
  4. May 20, 2012 #3
    Three equations, three unknowns... a bit of devil to solve easily though.

    For the first equation in αβγ space, I'm seeing four hyperbolic surfaces in +++ and three +-- octants
    The second equation a sphere of course - this tells me the magnitudes of αβγ are all less than √20 < 4.5
    The third equation therefore limits me to the +++ case (121 is nearly 5^3)...
  5. May 20, 2012 #4


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    You are given what α222 is. So what happens if you consider what (α+β+γ)2 is? (expand it out and see what terms you have)
  6. May 20, 2012 #5

    I like Serena

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    Hi songoku! :smile:

    Related to Vieta's formulas are Newton's identities.
    They work out the same as rock.freak667's suggestion.

    From these you can get a relation between your unknown coefficients and the equations that you are given.
    You won't find nice round numbers though.
  7. May 20, 2012 #6
    There are some posts above that I don't actually understand but let me try

    let : α+β+γ = p ; αβ + αγ + βγ = q

    (α+β+γ)2 = α2 + β2 + γ2 + 2 (αβ + αγ + βγ)
    p2 = 20 + 2q
    q = 1/2 (p2 - 20)

    (α+β+γ)3 = α3 + β3 + γ3 + 3(α+β+γ)(αβ + αγ + βγ) - 3 αβγ
    p3 = 121 + 3pq - 18

    subs. q to second equation results in cubic equation in terms of p, then by using calculator I got p = -6.7

    Am I correct? How to find p manually?

    Last edited: May 21, 2012
  8. May 21, 2012 #7


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    hi songoku! :smile:

    your method looks fine

    there's no easy way to solve a cubic polynomial (unless you know the roots are integers) :redface:
  9. May 21, 2012 #8
    hi tiny-tim :smile:

    *sigh...* hope that the root will be integer when exam comes....

    thanks a lot for the help :smile:
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