How to find number of zeros of complex polynomial in some area?

Question

anonymous · Answer

umm if you can post a problem, I might be able to help you.

gg · Answer

ok, wait a second. thanks :)

gg · Answer

how many zeros polynomial  P(z)= 3z^6 +z^5+ z^4+ 2z^3 + 10z^2+ z+1 has in area 1<|z|<2

anonymous · Answer

potential zeros are

$$\pm1,1/3$$

anonymous · Answer

oh, since the signs does not change anywhere, I think there are no real solutions

anonymous · Answer

Yup, but what about Complex...

anonymous · Answer

all 6 roots are complex

anonymous · Answer

There are complex solutions of course http://www.wolframalpha.com/input/?i=3z^6+%2Bz^5%2B+z^4%2B+2z^3+%2B+10z^2%2B+z%2B1

gg · Answer

thanks, but I need to know HOW to find zeros in that area

gg · Answer

I can't use wolfram on exam :D

anonymous · Answer

Me too...

anonymous · Answer

1<|z|<2 represent range of real number(I believe) but there are no real roots

anonymous · Answer

I know you can't :)
I did this kinds stuff over 2 decades ago and haven't touched it since - and don't remember much of it.

anonymous · Answer

Imran, why does  1<|z|<=2  represent real numbers ?

anonymous · Answer

nvm, I didn't read the question, they meant on complex plane

anonymous · Answer

$$|1+i|=\sqrt{2}$$

anonymous · Answer

Gg  - any clues you can give us from notes you took in class ?
:)

anonymous · Answer

You can try the factors of the last term divided by the factors of the first term. That's how imranmeah probably got the ±1, ±1/3, and then try synthetic division to see if any of them work...