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mr fantastic · #2 September 8th, 2008, 12:03 AM

Quote:

Originally Posted by sqleung

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:

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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 $P(x)$

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

Thank you.

I think more information is needed.

The following users thank mr fantastic for this useful post:
sqleung
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