Complex Polynomial of nth degree

[Nathew]

Homework Statement

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

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

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?