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johnny December 17th, 2007 08:56 PM

Fundamental Theorem of Algebra
 
Can anyone give me a short simple definition of FTOA?

cknapp December 18th, 2007 01:44 AM

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.

johnny December 18th, 2007 06:50 PM

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...?

cknapp December 18th, 2007 07:50 PM

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.

johnny December 18th, 2007 08:29 PM

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?

cknapp December 19th, 2007 12:22 AM

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

CRGreathouse December 19th, 2007 08:46 AM

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.

skipjack December 21st, 2007 12:54 AM

The statement of the theorem should specify "if repeated roots are counted up to their multiplicity".

cknapp December 21st, 2007 09:25 AM

well noted, skipjack.

johnny December 21st, 2007 12:58 PM

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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