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Old 09-07-2008, 11:56 PM
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Default Abstract Polynomials Question

Hello. I seem to be having a little trouble with this problem regarding quartics so if you could assist me, it would be greatly appreciated:

============================

A polynomial with real coefficients and two integer zeroes p and q is given as:

P(x) = x^4 + ax^3 + bx^2 + cx - 10

P(x) has a complex zeroes 1 + ki and 1 - ki

\bullet Using p and q, write another expression for a real quadratic factor of P(x) and using this, list the possible values of pq whereby p, q and 1 + ki are the zeroes of P(x)

So far, using sum and product, I wrote it as a quadratic factor:

x^2 - (p + q)x + pq

That means the quartic can be written as (using the real and complex factor):

P(x) = [x^2 - 2x + (1 + k^2)][x^2 - (p + q)x + pq]

Is this right so far?

\bullet Given p + q = -1, show that there's only one possible value for pq and hence, find all the zeroes of P(x)

============================

If you could help me out here, it would be greatly appreciated.

Thank you.
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  #2  
Old 09-08-2008, 12:03 AM
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Originally Posted by sqleung View Post
Hello. I seem to be having a little trouble with this problem regarding quartics so if you could assist me, it would be greatly appreciated:

============================

A polynomial with real coefficients and two integer zeroes p and q is given as:

P(x) = x^4 + ax^3 + bx^2 + cx - 10

P(x) has a complex zeroes 1 + ki and 1 - ki

\bullet Using p and q, write another expression for a real quadratic factor of P(x) and using this, list the possible values of pq whereby p, q and 1 + ki are the zeroes of P(x)

So far, using sum and product, I wrote it as a quadratic factor:

x^2 - (p + q)x + pq

That means the quartic can be written as (using the real and complex factor):

P(x) = [x^2 - 2x + (1 + k^2)][x^2 - (p + q)x + pq]

Is this right so far?

Mr F says: Yes. Note that the answer to the question is therefore {\color{blue}-10 = (1 + k^2) p q}.

\bullet Given p + q = -1, show that there's only one possible value for pq and hence, find all the zeroes of P(x)

============================

If you could help me out here, it would be greatly appreciated.

Thank you.
I think more information is needed.
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  #3  
Old 09-08-2008, 12:28 AM
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This was all the information I was provided with . You mean there's no other way to find the zeroes of the quartic?

Thanks though. Your help was definitely appreciated.
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Old 09-08-2008, 12:41 AM
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This was all the information I was provided with . You mean there's no other way to find the zeroes of the quartic?

Thanks though. Your help was definitely appreciated.
Are you meant to get them in terms of a, b and c?
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Old 09-08-2008, 12:48 AM
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Originally Posted by mr fantastic View Post
Are you meant to get them in terms of a, b and c?
It doesn't state but I guess that's what you're supposed to do.
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