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Homework Help: Quartic roots questions

  1. Mar 9, 2010 #1
    1. The problem statement, all variables and given/known data
    The equation x4-6x3-73x2+kx+m=0 has two positive roots, [tex]\alpha[/tex] [tex]\beta[/tex], and two negative roots, [tex]\delta[/tex] [tex]\gamma[/tex]. It is given that [tex]\alpha[/tex][tex]\beta[/tex]=[tex]\delta[/tex][tex]\gamma[/tex]=4

    (i) Find the values of k and m

    2. Relevant equations
    stated above


    3. The attempt at a solution

    m = ([tex]\alpha[/tex][tex]\beta[/tex])([tex]\gamma[/tex][tex]\delta[/tex])=4x4=16

    not too sure how to approach solving k
     
  2. jcsd
  3. Mar 9, 2010 #2

    epenguin

    User Avatar
    Homework Helper
    Gold Member

    You can at least take the natural step and then consider.

    You got your result by comparing m with the constant term in the expansion of


    [tex](x - \alpha[/tex])([tex]x - \beta)(x-\gamma[/tex])([tex]x - \delta[/tex]) .

    So you should be able to write what coefficient of [tex]x[/tex] is in the expansion (+ has probably been covered in your course) so at least write that down and see if there is anything you can do with it.

    (As far as I can see this comes out a whole number, but the equation itself does not have any nice whole number solutions.)
     
  4. Mar 11, 2010 #3
    It seems that you are confused, I'm giving the relations between the roots and coefficients in such equation

    First make the coefficient of highest term as unity, in your case it is already one.

    Now,

    (1). Second term coefficient is equal to the sum of roots but with its sign changed.

    Note: For simplicity I'm writing roots as A, B, C and D

    so here, A + B + C + D = -(-6)
    = 6 ..........................(equation 1)

    (2) Third coefficient is equal to the product of roots taken two at the time

    so here AB + BC + CD + DA = -73 ........................ (No use for you in this problem)

    (3) Fourth Coefficient (which you have to find "k") is equal to the product of roots taken three at the time with its sign changed.

    so here

    ABC + BCD + CDA + DAB = - k
    Given AB = CD = 4

    4C + 4B + 4A + 4D = -k

    4 (A + B + C + D) = -k

    From equation (1)

    4 (6) = -k

    k = -24 ................. (your first answer)

    (4) and last coefficient (constant therm here ) is equal to the sum of roots taken four at a time

    so here

    ABCD = m

    given AB = CD =4

    so 4 (4) = m

    m = 16........................ (your second answer)
     
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