Results 1 to 3 of 3

Math Help - Elementary Functions and Polynomials

  1. #1
    Newbie
    Joined
    May 2011
    Posts
    3

    Elementary Functions and Polynomials

    Let α, β, γ and δ denote the zeros of the quartic polynomial, f, given by
    x^4 + 5x^3 − x^2 + 3x + 1.

    i) Write down the four expressions for the elementary symmetric functions in α, β, γ and δ and give their values for this quartic polynomial.

    ii) Express the symmetric functions α^2 + β^2 + γ^2 + δ^2 and Σ(α − β)^2
    (six terms) in terms of these elementary symmetric functions and hence calculate their exact values. Calculate the exact value of α^(−3) + β^(−3) + γ^(−3) + δ^(−3).

    So I know that the elementary functions are as follows:
    Σα = α + β + γ + δ = -5
    Σαβ = αβ + αγ + αδ + βγ + βδ + γδ = -1
    Σαβγ = αβγ + αβδ + αγδ + βγδ = -3
    αβγδ = 1

    How do i proceed to do part (ii)? I am confused and the way my professor did it confused me even more. So can someone explain in detail?

    Thanks in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Mar 2010
    Posts
    715
    Thanks
    2
    a^2+b^2+c^2+d^2 = (a+b+c+d)^2-2(ab+ac+bc+ad+bd+cd).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,401
    Thanks
    762
    (a-b)^2 + (a-c)^2 + (a-d)^2 + (b-c)^2 + (b-d)^2 + (c-d)^2

    = a^2 - 2ab + b^2 + a^2 - 2ac + d^2 + a^2 - 2ad + d^2 + b^2 - 2bc

    + c^2 + b^2 - 2bd + d^2 + c^2 - 2cd + d^2

     = 3(a^2 + b^2 + c^2 + d^2) - 2(ab + ac + ad + bc + bd + cd)

    now use TheCoffeeMachine's result above to finish.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Elementary Functions Logarithm question
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: June 29th 2011, 05:54 PM
  2. Integral in terms of elementary functions
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 18th 2011, 08:58 PM
  3. Inverse Functions of polynomials
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: February 6th 2010, 10:23 AM
  4. Elementary symmetric polynomials
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: August 22nd 2008, 06:36 AM
  5. Elementary Functions as a Solvable Group
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: July 13th 2007, 01:19 PM

Search Tags


/mathhelpforum @mathhelpforum