Roots of a polynomial

**zadir** · March 26th 2011, 01:38 PM

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!

**ojones** · March 27th 2011, 05:18 PM

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.

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