[SOLVED] Complex factors of polynomials

Quote:

Originally Posted by Evales

Question:
Find real values for k for which (z - ki) is a factor of the polynomial equation: $z^4-2z^3+7z^2-4z+10=0$

What I did:
I tried splitting them up into two different quadratic equations to use the Qudaratic formula to find the complex roots. But I ended up with a mess of unknowns and it just seemed to complicated and messy to be the right direction.

Any hints as to how I should tackle these questions?

Let $p(z) = z^4-2z^3+7z^2-4z+10$ . You require p(ki) = 0:

$(ki)^4-2(ki)^3+7(ki)^2-4(ki)+10 = k^4 + 2k^3 i - 7k^2 - 4ki + 10 = (k^4 - 7k^2 + 10) + i (2k^3 - 4k)$ .

Therefore you require values of k that simultaneously satisfy:

$k^4 - 7k^2 + 10 = 0$ .... (1)

$2 k^3 - 4k = 0 \Rightarrow k^3 - 2k = 0$ .... (2)

Solutions to (2) are $k = 0, \, k = \sqrt{2}, \, k = -\sqrt{2}$ .

Keep the value(s) of k that satisfy (1) .....