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Math Help - Finding Linear Factors Over C

  1. #1
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    Finding Linear Factors Over C

    Hi, the problem I have is the following:

    z^2 + 4z + 10, finding the factors over C.

    I have gotten up to: (-2 + (sqrt6)i) and (-2 - (sqrt6)i).

    I also know, however, that the answers should be

    (z + 2 + (sqrt6)i)(z + 2 - (sqrt6)i).

    The problem is i can't see how i can change my original answer to this one. What do I need to do?

    Thanks
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Stevo_Evo_22 View Post
    Hi, the problem I have is the following:

    z^2 + 4z + 10, finding the factors over C.

    I have gotten up to: (-2 + (sqrt6)i) and (-2 - (sqrt6)i).

    I also know, however, that the answers should be

    (z + 2 + (sqrt6)i)(z + 2 - (sqrt6)i).

    The problem is i can't see how i can change my original answer to this one. What do I need to do?

    Thanks
    z^2+4z+10,~~F_1,F_2\in\mathbb{C}

    (where F_1 and F_2 are complex factors of z)

    So when you solved for z, I'm sure you saw that z=\frac{-4\pm\sqrt{16-40}}{2}\implies z=-2\pm\sqrt{6}i

    note that these are zeros.

    However, to get factors, we want to get the expressions F_1=0 and F_2=0, so we see that z+2-\sqrt{6}i=0 and z+2+\sqrt{6}i=0

    So we see that z^2+4z+10=(z+2+\sqrt{6}i)(z+2-\sqrt{6}i)

    I hope this is clear enough. If you have any additional questions, feel free to ask!

    --Chris
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  3. #3
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    Quote Originally Posted by Stevo_Evo_22 View Post
    Hi, the problem I have is the following:

    z^2 + 4z + 10, finding the factors over C.

    I have gotten up to: (-2 + (sqrt6)i) and (-2 - (sqrt6)i).

    I also know, however, that the answers should be

    (z + 2 + (sqrt6)i)(z + 2 - (sqrt6)i).

    The problem is i can't see how i can change my original answer to this one. What do I need to do?

    Thanks
    z^2+4z+10

    you got (-2- i \sqrt{6}) \;and\; (-2+ i \sqrt{6})

    Now the factors are

    [z-(-2- i \sqrt{6})].[z-(-2+ i \sqrt{6})]

    [z+2 + i \sqrt{6})].[z+2- i \sqrt{6})]
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  4. #4
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    Thank you very much. At the start I wasn't sure exactly what i was finding, so knowing that they were possible solutions of z, it's cleared it up a lot
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