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Zeros of a polynomial

  1. May 17, 2008 #1
    [SOLVED] zeros of a polynomial

    1. The problem statement, all variables and given/known data
    Find the zeros of the polynomial

    [tex]P(x) = x^4-6x^3+18x^2-30x+25[/tex]

    knowing that the sum of two of them is 4.


    2. Relevant equations
    http://en.wikipedia.org/wiki/Viète's_formulas


    3. The attempt at a solution
    Let x_1,x_2,x_3,x_4 be the complex roots and let x_1 +x_2 = 4. Here are the Viete relations in this case:

    [tex]x_1+x_2+x_3+x_4 = 6 [/tex]

    [tex] x_1 x_2 +x_1 x_3 +x_1 x_4 + x_2 x_3 + x_2 x_4 +x_3 x_4 = 18 [/tex]

    [tex] x_1 x_2 x_3 + x_1 x_3 x_4 +x_2 x_3 x_4 +x_1 x_2 x_4= 30 [/tex]

    [tex] x_1 x_2 x_3 x_4 = 25[/tex]

    The first one implies that x_3 +x_4 =2. And then the second one implies that x_1 x_2 + x_3 x_4 = 10 but that is as far as I can get.

    Please just give a hint.
     
  2. jcsd
  3. May 17, 2008 #2

    tiny-tim

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    Hint: so x_2 = 4 - x_1 and x_4 = 2 - x_3.

    Go substitute! :smile:
     
  4. May 17, 2008 #3
    Do you mean into

    [tex]
    P(x) = x^4-6x^3+18x^2-30x+25
    [/tex]

    ?
     
  5. May 17, 2008 #4

    tiny-tim

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    No … I mean substitute into your Viete relations. :smile:
     
  6. May 17, 2008 #5
    Sorry tiny-tim, I don't know see why that helps:

    For the second Viete relation I get:

    [tex]4x_1 - 2 x_3 - x_1^2-x_3 ^2+ x_1 x_3 = 10[/tex]

    For the third one I get

    [tex]8x_1 + 8x_3 - 2 x_1 ^2 - 4 x_3^2 = 30[/tex]

    For the fourth one I get

    [tex]8 x_1 x_3 + x_1 ^2 x_ 3^2 - 2x_1 ^2 x_3 - 4 x_1 x_3 ^2 = 25[/tex]

    Is there a simple way to solve these equations?
     
  7. May 17, 2008 #6

    tiny-tim

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    hmm … turned out more complicated than I thought. :frown:

    Well … that's what happens in exams sometimes … you try something, and it doesn't work, so you try the next most obvious thing … :smile:

    Now this will work:

    in your
    put (x_1 + x_2)s together, and (x_3 + x_4)s, and you should get two equations in x_1x_2 and x_3x_4, from which you get x_1x_2 = … and x_3x_4 = … ? :smile:
     
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