[SOLVED] Complex factors of polynomials

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

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?

Quote:

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Originally Posted by Evales

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

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?

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

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

Therefore you require values of k that simultaneously satisfy:

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

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

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

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