# Thread: Roots of a polynomial

1. ## 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!

2. This result is proved using the triangle inequality:

$
|x|-|y|\le |x+y|\le |x|+|y|.
$

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

$
0= |p({\alpha)|\ge |\alpha|^n-(|\alpha|^{n-1}+\cdots |\alpha}|+1).
$

The result follows quickly from this.