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July 15th, 2009, 06:55 PM  #1 
Newbie Joined: Jul 2009 Posts: 6 Thanks: 0  Show that a polynomial has a zero outside unit circle
Hi, I'm stuck in proving the following: Show that if p(z) = z^n + a_n1*z^n1 + ... + a_0 is a polynomial of degree n ? 1 and a_0 > 1, then p(z) has at least one zero outside the unit circle. Notice that the leading coefficient a_n = 1. There is also a hint given: Consider the factored form of p(z). This problem is from the book Fundamentals of Complex Analysis by Saff and Snider. 
July 16th, 2009, 03:01 AM  #2 
Newbie Joined: Jun 2009 Posts: 20 Thanks: 0  Re: Show that a polynomial has a zero outside unit circle
Consider . So we have . Notice that multiplying numbers in the unit circle gives another unit. So if were units then would also be a unit. But given is 
July 16th, 2009, 07:10 AM  #3 
Newbie Joined: Jul 2009 Posts: 6 Thanks: 0  Re: Show that a polynomial has a zero outside unit circle
That was quite simple. Thank you very much for you answer.

July 16th, 2009, 12:51 PM  #4  
Newbie Joined: Jun 2009 Posts: 20 Thanks: 0  Re: Show that a polynomial has a zero outside unit circle
No problem Quote:
 
August 4th, 2009, 10:59 PM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Show that a polynomial has a zero outside unit circle
Equations have roots and expressions (e.g. polynomials) have zeros.

August 5th, 2009, 03:32 PM  #6 
Newbie Joined: Jun 2009 Posts: 20 Thanks: 0  Re: Show that a polynomial has a zero outside unit circle
Guess I was wrong then :}

December 3rd, 2009, 06:25 PM  #7  
Senior Member Joined: Dec 2009 Posts: 150 Thanks: 0  Re: Show that a polynomial has a zero outside unit circle Quote:
 

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