1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    α222=20
    α333=121

    Find the equation of cubic polynomial


    2. Relevant equations
    vieta


    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

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    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

    rock.freak667

    User Avatar
    Homework Helper

    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

    User Avatar
    Homework Helper

    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?

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

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    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:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Find the equation of cubic polynomial
  1. Cubic Polynomials (Replies: 3)

Loading...