Complex Analysis Final Tomorrow

[hypn0tik]

Anyone know how to do the following problem?

How many zeros does z^7 + 4z^4 + z^3 + 1 have in each of the regions |z|<1 and 1 < |z| < 2?

How many zeros of this polynomial are in the first quadrant?

Any ideas on how to start? I think I need to use Rouche's Theorem and/or argument of the principle, but I'm not exactly sure.

Any experts?

 

[hypn0tik]

Only 3 views?

 

[tfinch2]

42

 

[PurdueRy]

factor it and then find the zeros. you will likely find that some are complex. The ones that are complex are the ones that can be in different quadrants using the standard rectangular system

 

[Random Variable]

If I take a class in complex analysis, it will be next fall. This semester I took advanced vector calculus. Next semester I signed up for real analysis, a class that I tried a year ago for only 2 weeks before I wanted to shoot myself in my face.

 

[hypn0tik]

Originally posted by: PurdueRy
factor it and then find the zeros. you will likely find that some are complex. The ones that are complex are the ones that can be in different quadrants using the standard rectangular system

Factoring would be a pain in the ass. I don't think factoring is the answer there.

 

[PurdueRy]

Originally posted by: hypn0tik

Originally posted by: PurdueRy
factor it and then find the zeros. you will likely find that some are complex. The ones that are complex are the ones that can be in different quadrants using the standard rectangular system

Factoring would be a pain in the ass. I don't think factoring is the answer there.

why is factoring so hard??

 

[Mday]

Originally posted by: hypn0tik
Anyone know how to do the following problem?

How many zeros does z^7 + 4z^4 + z^3 + 1 have in each of the regions |z|<1 and 1 < |z| < 2?

How many zeros of this polynomial are in the first quadrant?

Any ideas on how to start? I think I need to use Rouche's Theorem and/or argument of the principle, but I'm not exactly sure.

Any experts?

been a few years since i did this, was an easy A though...

I suggest against working with Rouche's theorem unless you know what you are doing. the contours are going to hurt you in this problem. Rouche's theorem forces you to work with another function that would cause problems if you are not experienced.

argument principle:
http://mathworld.wolfram.com/ArgumentPrinciple.html
you know there are no poles since the function f(z) you have is entire\analytic in C (duh its a polynomial). Furthermore it's also a meromorphic function, i'll leave it to you to work with that. your 2 and 1 circles define the contours since you're given a strict inequality.

so all you gotta do is integrate using the argument principle.

GL HF.