Roots of a Polynomial

  1. dbr

    dbr 1

    1. The problem statement, all variables and given/known data

    Let [itex]z^n + \sum_{k=0}^{n-1}a_kz^k[/itex] be a polynomial with real coefficients [itex]a_k\in[0,1][/itex]. If [itex]z_0[/itex] is a root, prove that [itex]Re(z_0) < 0[/itex] or [itex]|z_0| < \frac{1+\sqrt{5}}{2}[/itex].

    2. Relevant equations

    3. The attempt at a solution

    I have attempted to solve this problem by contradiction (i.e. assuming there is a root [itex]z_0[/itex] with [itex]Re(z_0) \geq 0[/itex] and [itex]|z_0| \geq \frac{1+\sqrt{5}}{2}[/itex]). I then tried to look for a contradiction in the equations [itex]Re(z_0^n + \sum_{k=0}^{n-1}a_kz_0^k) = 0[/itex] and [itex]Im(z_0^n + \sum_{k=0}^{n-1}a_kz_0^k) = 0[/itex]. Unfortunately, I'm not able to find any contradiction.
  2. jcsd
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