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Math Help - Complex numbers question

  1. #1
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    Complex numbers question

    Firstly, I apologise if this is in the wrong forum. I spent quite a bit of time looking through the different forums and this seems to be the one most appropriate.

    Let \alpha be such that \alpha^3 = -1 and \alpha\neq-1. Evaluate (\alpha^2(\alpha - 1)^2)^{-1} without calculating explicitly the values of \alpha.

    So far I have rewritten (\alpha^2(\alpha - 1)^2)^{-1} as (\alpha^2-\alpha+2)^{-1} by substituting \alpha^3 = -1 but am not sure how to proceed from here.
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  2. #2
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    Re: Complex numbers question

    Yes, you are correct that \alpha^2(\alpha- 1)^2= \alpha^2(\alpha^2- 2\alpha+ 1) = \alpha^4- 2\alpha^3+ \alpha^2= -\alpha+ 2+ \alpha^2 because \alpha^3= 1.

    The fact that \alpha= -1 satisfies \alpha^3= -1 means that \alpha^3+ 1 factors:
    \alpha^3+ 1= (\alpha+ 1)(\alpha^2- \alpha+ 1). Here, since \alpha^3+ 1= 0 but \alpha\ne 0, we have \alpha^2- \alpha+ 1= 0.

    Finally, that \alpha^2- \alpha + 2= (\alpha^2- \alpha+ 1)+ 1. Easy to find the reciprocal of that, isn't it?
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