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Math Help - [SOLVED] Complex factors of polynomials

  1. #1
    Junior Member Evales's Avatar
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    [SOLVED] Complex factors of polynomials

    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?
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  2. #2
    Flow Master
    mr fantastic's Avatar
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    Quote Originally Posted by Evales View Post
    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) .....
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