Fundamental Theorem of Algebra Proof

riskybeats

Hello, trying to figure out exactly what is going on in this question.

1. The problem statement, all variables and given/known data

(a) If P(z) is a nonconstant polynomial, show that |P(z)| > |P(0)| holds outside
some disk R |z| ≤ R for some R > 0. Conclude that if the minimum value of |P(z)| for R
z ≤ |R| occurs at z_o, then z=z_o gives the minimum value of |P(z)| with respect to the whole complex plane.

2. Relevant equations

3. The attempt at a solution

P(z) = a0+a1*z1+a2*z^2+...+an*zn^n
|P(z)| = |a0+a1*z1+a2*z^2+...+an*zn^n|
|P(z)| = |Z**n||a0*z^-n+a1*z1^1-n+a2*z^2-n+...+an|

I know that I have to show that P(z) grows faster than P(0) for it to hold outside the disk. But I am not sure what that even means for an inequality to hold outside of a disk. I really don't know what it is asking.

HallsofIvy

Well, where, exactly, is your difficulty? Do you know what a "disk" is? Do you know what is meant by "outside of a disk"? Do you know what it means for "an inequality to hold"?