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December 17th, 2007, 07:56 PM  #1 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0  Fundamental Theorem of Algebra
Can anyone give me a short simple definition of FTOA?

December 18th, 2007, 12:44 AM  #2 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 
Every complex polynomial of degree n has exactly n complex roots. if y= ax^n + bx^(n1) + ... z, where a,b,...,z are complex (which includes the reals), y will have n roots (where y =0), each of which is complex. 
December 18th, 2007, 05:50 PM  #3  
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0  Quote:
For example, if (23i)x^5  (3+6i)x^3 + (2i)x^2 + i = 0, then there are 5 roots, which are complex? And, if 2x^5  (3i)x^4 + (22i)x^3  6x^2  2 = 0, then there are 5 roots, which are complex...?  
December 18th, 2007, 06:50 PM  #4 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 
The real numbers are a subset of the complex number system. So any real polynomial (e.g. x^2 + 2x + 1) is a special case of the complex number system. So in answer to your question, it is not necessary that it is not real, just that it is complex. Think of reals as complex numbers where the b in a+bi is 0. a+0i=a. 
December 18th, 2007, 07:29 PM  #5 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0 
Oh yeah, I remember now that a+bi is complex, where not necessary imaginary, because b=0. So, using this knowledge, the Fundamental Theorem of Algebra tells us that if we have a polynomial equal to zero, and its degree n is greater then 0, then it has n roots. Is this correct?

December 18th, 2007, 11:22 PM  #6 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 
yes. The root of a polynomial is when the polynomial is equal to zero. So there are n values for which the polynomial is 0

December 19th, 2007, 07:46 AM  #7  
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  Quote:
a + 0i is real. 0 + bi is imaginary. The only imaginary real number is 0.  
December 20th, 2007, 11:54 PM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,173 Thanks: 1644 
The statement of the theorem should specify "if repeated roots are counted up to their multiplicity".

December 21st, 2007, 08:25 AM  #9 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 
well noted, skipjack.

December 21st, 2007, 11:58 AM  #10  
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0  Quote:
 

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