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November 3rd, 2012, 10:48 AM   #1
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polynomial

I would like to know the real and imaginary solutions of a polynomial with higher degree... how can I do that? I am not interested which are them but how many real and how many imaginary there are, that's all. :P
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November 3rd, 2012, 11:14 AM   #2
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Re: polynomial

The number of solutions (complex & real) of a polynomial of degree n is <= n.
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November 3rd, 2012, 11:27 AM   #3
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Re: polynomial

Fundamental theorem of algebra. The proof is rather hard, though.
(There are two proofs that I know : one by the galois theory and another by the complex analysis.)
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November 3rd, 2012, 12:23 PM   #4
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Re: polynomial

The number of solutions (complex & real) of a polynomial of degree n is <= n - but how many of them are real and how many are imaginary? Thanks.
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November 3rd, 2012, 12:31 PM   #5
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Re: polynomial

That depends on the polynomial. Complex roots always come in conjugate pairs, so you know there will be an even number of complex roots. Other than that, you determine the number of complex and real roots on a case-by-case basis.
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November 3rd, 2012, 06:40 PM   #6
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Re: polynomial

Quote:
Originally Posted by alexmath
The number of solutions (complex & real) of a polynomial of degree n is <= n - but how many of them are real and how many are imaginary? Thanks.
You will first have to differentiate between a real polynomial and complex polynomial :

*Real polynomial : all coefficients are real , the complex roots come in conjugate pairs so :

1-if the polynomial is of degree n where n is even(probable solutions) :

a - the roots are all real ,
b - the roots are all complex conjugate pairs.
c - k roots are real and j roots are complex (where k and j are even)
*Note:if j/2 roots are distinct complex roots then the rest j/2 roots are complex conjugates.
d - It is impossible to have an odd number of real or complex roots.

2 - if the polynomial of degree n where n is odd (probable solutions):

a - all roots are real.
b - an odd number of roots is real ,the rest will be complex conjugate pairs.
c - it is impossible to have all roots are complex.
d - it is impossible to exist an even number of real roots.

*complex polynomials : (I don't have that much information you can search the web)

*conjugate pairs (if a+ib is a solution then a-ib is also a solution great property of real polynomials)
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