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Math Help - Complex polynomial

  1. #1
    Junior Member utopiaNow's Avatar
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    Complex polynomial

    Show that if p_n(z) = a_nz^n + a_{n-1}z^{n-1} + \cdots + a_1z + 1 is a polynomial of degree n with constant term 1, and if |a_n|  > 1, then at least one of the roots of p_n(z) lies inside the unit circle.

    My attempt: The hint that's been given is to use the "factored form", which I assume means p_n(z) = a_n(z - z_1)(z - z_2)\cdots(z - z_n).

    But I'm not sure how to advance from here. Any suggestions would be appreciated as I'm literally stuck on the first step.

    Edit: Never mind, I solved it.
    Last edited by utopiaNow; October 9th 2009 at 09:43 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by utopiaNow View Post
    Show that if p_n(z) = a_nz^n + a_{n-1}z^{n-1} + \cdots + a_1z + 1 is a polynomial of degree n with constant term 1, and if |a_n|  > 1, then at least one of the roots of p_n(z) lies inside the unit circle.

    My attempt: The hint that's been given is to use the "factored form", which I assume means p_n(z) = a_n(z - z_1)(z - z_2)\cdots(z - z_n).

    But I'm not sure how to advance from here. Any suggestions would be appreciated as I'm literally stuck on the first step.

    Edit: Never mind, I solved it.
    You have:

    a_n\prod_{i=1}^nz_i=1

    So:

    \prod_{i=1}^n|z_i|<1

    etc

    CB
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