Complex Polynomial of nth degree

[Nathew]

Homework Statement

Show that if
$$P(z)=a_0+a_1z+\cdots+a_nz^n$$
is a polynomial of degree $n$ where $n\geq1$ then there exists some positive number $R$ such that
$$|P(z)|>\frac{|a_n||z|^n}{2}$$
for each value of $z$ such that $|z|>R$

Homework Equations

Not sure.

The Attempt at a Solution

I've tried dividing through by the nth power of z. That way I can somehow incorporate the R value somehow but I'm not exactly sure where to go from here.

Thanks!

 

[PeroK]

Homework Statement

Show that if
$$P(z)=a_0+a_1z+\cdots+a_nz^n$$
is a polynomial of degree $n$ where $n\geq1$ then there exists some positive number $R$ such that
$$|P(z)|>\frac{|a_n||z|^n}{2}$$
for each value of $z$ such that $|z|>R$

Homework Equations

Not sure.

The Attempt at a Solution

I've tried dividing through by the nth power of z. That way I can somehow incorporate the R value somehow but I'm not exactly sure where to go from here.

Thanks!

Maybe you could start by showing that for large enough z,

$|z|^n > |a_0|$

And, perhaps, rewrite the equation with everything but $a_n z^n$ on the LHS.

Can you see, without doing any algebra, why it's true?