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!
This result is proved using the triangle inequality:
$\displaystyle
|x|-|y|\le |x+y|\le |x|+|y|.
$
Let $\displaystyle p(x)=x^n+a_{n-1}x^{n-1}+\cdots +a_0$ be the monic polynomial and $\displaystyle \alpha$ a root. Then,
$\displaystyle
0= |p({\alpha)|\ge |\alpha|^n-(|\alpha|^{n-1}+\cdots |\alpha}|+1).
$
The result follows quickly from this.