Thread: Complex Numbers help

Originally Posted by kmalik001

Thank you so much!
I understand it.

I am also kind of stumped on this question.

Determine the only real values a, b, c and d such that the equation:
z^4 + az^3 + bz^2 + cz + d = 0
has both z1 and z2 as roots.

I don't really get what this is asking.

Good!

I guess this requires a bit of additional knowledge about polynomial equations with real coefficients and complex roots.

A polynomial that has both z1 and z2 as its roots, must be of the form:

(z-z1)(z-z2)(...) = 0

If such an equation has only real coefficients, that implies that not only z1 must be a root, but also its conjugate z1*.
(I suspect that is somewhere in your notes.)

In other words, your equation must be of the form:

$\displaystyle (z-z_1)(z-z_1^*)(z-z_2)(z-z_2^*) = 0$

If you expand this, which coefficients do you get?

Originally Posted by Plato

@ ILikeSerena
I don't care that you responded to this thread.

Ah well, that's okay, as long as you don't mind if I slice your comments to pieces (assuming you give me the opportunity as you did this time round).

Ah yes I see what you are doing. So when I expanded it I get z^4 + 4z^3 + 0z^2 -200z + 500. Which I hope is right.
I expanded (z^2 -6z + 10)(z^2 +10 z +50), which I know are right since their solutions are the z1, z2 and their conjugates.

Thanks to both of you once again

Originally Posted by kmalik001

Ah yes I see what you are doing. So when I expanded it I get z^4 + 4z^3 + 0z^2 -200z + 500. Which I hope is right.
I expanded (z^2 -6z + 10)(z^2 +10 z +50), which I know are right since their solutions are the z1, z2 and their conjugates.

Thanks to both of you once again

Yep. That is right!

So there is another complex number question.

u = −1 + j√3 and v = √3 − j

Let a be a real scaling factor. Determine the value(s) of a such that
|u −a/v | = 2√2

So this is what I am doing I am kind of stuck and wondering if I am on the right track.