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Math Help - Roots of a polynomial

  1. #1
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    Roots of a polynomial

    Prove that if the absolute value of the coefficients of a complex polynomial with a leading coefficient of 1 is maximum 1, than the absolute values of its roots are smaller than 2.

    Any help would be appreciated!
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  2. #2
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    This result is proved using the triangle inequality:

    <br />
|x|-|y|\le |x+y|\le |x|+|y|.<br />

    Let p(x)=x^n+a_{n-1}x^{n-1}+\cdots +a_0 be the monic polynomial and \alpha a root. Then,

    <br />
0= |p({\alpha)|\ge |\alpha|^n-(|\alpha|^{n-1}+\cdots |\alpha}|+1).<br />

    The result follows quickly from this.
    Last edited by ojones; March 27th 2011 at 07:15 PM.
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