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Crafting an equation using roots of another equation

  1. Jul 30, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the equation whose roots are given by adding 2 to the roots of the equation
    x^4+3x^3-13x^2-51x-36=0


    2. Relevant equations
    x^4-(Σa)x³+(Σab)x²-(Σabc)x+abcd = 0

    3. The attempt at a solution

    And the coefficient of x^3 to be -5
    second coefficient = b/a = -Σa = 3
    Since we are adding two to every root so b/a = -(Σa +2) = -(Σa +Σ2) = -(-3+8) = -5 thus the equation should have x^4-5x³

    And I am stuck at x^2
    =Σ(a+2)(b+2)
    =Σab+2a+2b+4
    =Σab+ 4Σa +16 which amounts to -9
    Correct coefficient is -7

    Note:- We aren't allowed to solve the given equation in any part of the working
     
  2. jcsd
  3. Jul 30, 2016 #2

    LCKurtz

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    Hint: If ##r## is a root of ##f(x)## what would be a root of ##f(x-2)##?
     
  4. Jul 30, 2016 #3

    ehild

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    It is wrong. Show your working.
    The correct coefficient is really -7.

     
  5. Jul 31, 2016 #4
    r+2
     
  6. Jul 31, 2016 #5
    Ehild, I am aware the coefficient is wrong, the working I have given is my wrong working and I want to know where I went wrong
     
  7. Jul 31, 2016 #6

    haruspex

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    Right. Can you see how to use that to make a substitution for x in the polynomial?
     
  8. Jul 31, 2016 #7
    I am aware of the method you are implying, but the question forbids me to use that method.
     
  9. Jul 31, 2016 #8

    haruspex

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    No, it says
    which is not what I have in mind.
     
  10. Jul 31, 2016 #9

    ehild

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    Your notations are not clear. What do you mean with Σa?
    If x1, x2, x3 x4 are the roots of the original equation, the coefficient of the x2term is
    Σ(xi+2)(xj+2) for all pairs i<j: it is 6 terms as you have 6 pairs from 4 elements.
    So it is (x1x2+x1x3+x1x4+x2x3+x2x4+x3x4)+2(x1+x2+x1+x3+x1+x4+x2+x3+x2+x4+x3+x4)+6*4, as you have 6 times 2*2.
    Also you have 3Σxi in the parentheses of the second term, instead of 2.
     
    Last edited: Jul 31, 2016
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