A Polynomials question

sqleung · #1 $Old$ 09-05-2008, 02:05 AM

Cam someboy please help me out with this? I've done a part of it so far. This is the question:

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Given the polynomial $P(x) = x^4 + ax^3 + bx^2 + cx - 10$ has real coefficients.

IF we know $P(x)$ has a complex zero $1 + ki$ (k being an integer), use this to find an expression for a real quadratic factor of $P(x)$ .

Hence, find all the possible values for $k$ for which the quadratic is a factor of $P(x)$

Now, this is what I have done so far:

If $1 + ki$ is a zero, then $1 - ki$ is also a zero (conjugates).

From this, I used the sum and product rules:

SUM : $(1 + ki) + (1 - ki) = 2$
PRODUCT : $(1 + ki)(1 - ki) = 1 - k^2$

This gives $x^2 - 2x + (1 - k^2)$

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What do I do with this? Now that I have a quadratic factor, I'm supposed to find the possible values of k. I was thinking maybe $k > 0$ and that's the answer but I'm not sure (square rooting negative number gives an imaginary solution)

Can someone check it for me?

Thanks. All help is appreciated.

sqleung · #2 $Old$ 09-05-2008, 02:32 AM

Thanks!

That's just what I needed.

flyingsquirrel · #3 $Old$ 09-05-2008, 02:35 AM

[this post should be #2, I've deleted it for no reason]
Hello,

Quote:

Originally Posted by sqleung $View Post$

PRODUCT : $(1 + ki)(1 - ki) = 1 - k^2$

There is a typo (?) here : the product is $(1 + ki)(1 - ki) = 1 {\color{red}+} k^2$ .

Quote:

What do I do with this? Now that I have a quadratic factor, I'm supposed to find the possible values of k. I was thinking maybe $k > 0$ and that's the answer but I'm not sure (square rooting negative number gives an imaginary solution)

It seems good to me .

bobak · #4 $Old$ 09-05-2008, 03:59 PM

Did the give us all of the question? is it possible that you may have been told that the other roots are integers? in that case what are the possible values of k?

Bobak