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Help with solving a polynomial

  1. Oct 9, 2004 #1
    I need some help on how to solve this question. It asks me to find all real numbers a with the property that the polynomial equation x^10 + a*x +1 = 0 has a real solution r such that 1/r is also a solution. I tried plugging in r and 1/r and equating the 2 equations, but that got me nowhere.
     
    Last edited: Oct 9, 2004
  2. jcsd
  3. Oct 9, 2004 #2
    rearrange the equation as,
    a = (-x^10-1)/x
    put x=r and call it eqn 1
    then put x=1/r and call it eqn 2
    shouldn't RHS of both 1 and 2 be same?

    -- AI
     
  4. Oct 9, 2004 #3

    Hurkyl

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    Well, could you show what you got when you plugged in r and 1/r?
     
  5. Oct 9, 2004 #4
    i euqated that equation and found out that r= + or - 1 so a = + or - 2.
    well, that yielded 2 equations. x^10 - 2x + 1 and x^10 + 2x +1. r = 1 is a zeo, but r=-1 is not. what am i doing wrong?
     
  6. Oct 9, 2004 #5
    Do you know the answer?
     
  7. Oct 10, 2004 #6
    for r = 1, a=-2
    so x^10 - 2x + 1 = 0
    put r = 1 , it is zero .... put 1/r = 1 .. again it is zero.

    for r=-1, a = 2
    so x^10 + 2x + 1 = 0
    put r = -1 , it is zero .... put 1/r = -1 .. again it is zero.

    so our conditions are satisfied...

    -- AI
     
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