# Complex polynomial

• Oct 9th 2009, 09:12 AM
utopiaNow
Complex polynomial
Show that if \$\displaystyle 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 \$\displaystyle |a_n| > 1\$, then at least one of the roots of \$\displaystyle 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 \$\displaystyle 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.
• Oct 9th 2009, 10:52 AM
CaptainBlack
Quote:

Originally Posted by utopiaNow
Show that if \$\displaystyle 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 \$\displaystyle |a_n| > 1\$, then at least one of the roots of \$\displaystyle 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 \$\displaystyle 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:

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

So:

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

etc

CB