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Math Help - Complex numbers problem

  1. #1
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    Complex numbers problem

    Show that z^4 - 2z3 + 6z^2 -8z + 8 =0 has a root of the form ki, where k is real. Hence solve the equation z^4 - 2z3 + 6z^2 -8z + 8 =0

    my steps are
    let f(x) = z^4 - 2z3 + 6z^2 -8z + 8
    since, ki is a root
    f(ki)= z^4 - 2z3 + 6z^2 -8z + 8 = 0
    substituting ki inside,
    i get (k^2 -2)(k^2 -4) +2k(k^2 -4)i =0

    im not sure how to proceed at this point..

    the answer is z=2i, -2i, 1+i or 1-i
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  2. #2
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    Quote Originally Posted by mephisto50 View Post
    Show that z^4 - 2z3 + 6z^2 -8z + 8 =0 has a root of the form ki, where k is real. Hence solve the equation z^4 - 2z3 + 6z^2 -8z + 8 =0

    my steps are
    let f(x) = z^4 - 2z3 + 6z^2 -8z + 8
    since, ki is a root
    f(ki)= z^4 - 2z3 + 6z^2 -8z + 8 = 0
    substituting ki inside,
    i get (k^2 -2)(k^2 -4) +2k(k^2 -4)i =0 Notice that this factorises as \color{red}(k^2-4)(k^2-2+2ki)=0.

    im not sure how to proceed at this point..

    the answer is z=2i, -2i, 1+i or 1-i
    ..
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