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Roots of a polynomial of degree 4

  1. Nov 12, 2006 #1
    (*)[tex]p(x) = x^4 + ax^3 + bx^ 2 + ax + 1 = 0[/tex]

    where [tex]a,b \in \mathbb{C}[/tex]

    I would like to prove that a complex number x makes (*) true iff

    [tex]s = x + x^{-1}[/tex] is a root of the [tex]Q(s) = s^2 + as + (b-2) [/tex]

    I see that that [tex]Q(x + x^{-1}) = \frac{p(x)}{x^2}[/tex]

    Then to prove the above do I then show that p(x) and Q((x + ^{-1}) shares roots?

    Sincerely Yours
  2. jcsd
  3. Nov 13, 2006 #2


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    Yes. Note that Q(s) and p(x) must have coincident roots.
  4. Nov 13, 2006 #3


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    The onlly way a fraction can be 0 is if the numerator is 0. Isn't it obvious from [tex]Q(x + x^{-1}) = \frac{p(x)}{x^2}[/tex]
    that Q(x+ x-1)= 0 if and only if p(x)= 0?
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