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December 17th, 2007, 08:56 PM   #1
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Fundamental Theorem of Algebra

Can anyone give me a short simple definition of FTOA?
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December 18th, 2007, 01:44 AM   #2
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Every complex polynomial of degree n has exactly n complex roots.

if y= ax^n + bx^(n-1) + ... z, where a,b,...,z are complex (which includes the reals), y will have n roots (where y =0), each of which is complex.
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December 18th, 2007, 06:50 PM   #3
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Quote:
Originally Posted by cknapp
Every complex polynomial of degree n has exactly n complex roots.
So, all of the coefficients has to be complex number? And at the same time, n complex roots, where no real roots solution possible?


For example, if (2-3i)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 + (2-2i)x^3 - 6x^2 - 2 = 0, then there are 5 roots, which are complex...?
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December 18th, 2007, 07:50 PM   #4
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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.
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December 18th, 2007, 08:29 PM   #5
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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?
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December 19th, 2007, 12:22 AM   #6
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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
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December 19th, 2007, 08:46 AM   #7
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Quote:
Originally Posted by johnny
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?
a + bi is complex.

a + 0i is real.

0 + bi is imaginary.

The only imaginary real number is 0.
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December 21st, 2007, 12:54 AM   #8
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The statement of the theorem should specify "if repeated roots are counted up to their multiplicity".
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December 21st, 2007, 09:25 AM   #9
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well noted, skipjack.
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December 21st, 2007, 12:58 PM   #10
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Quote:
Originally Posted by skipjack
The statement of the theorem should specify "if repeated roots are counted up to their multiplicity".
What does this mean? What is multiplicity?
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