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Thread: De Moivre's Theorem of Complex numbers Question Help

  1. #1
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    Post De Moivre's Theorem of Complex numbers Question Help

    Hi I'm having trouble solving this complex number problem. If anyone can help me out i will really appreciate it. Thanks

    here is the question
    Use De Moivre's Theorem to solve this equation z9 + z5 - z4 -1 = 0?

    I just want to know how can i start solving this??
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  2. #2
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    Re: De Moivre's Theorem of Complex numbers Question Help

    The LHS will factorise. Then you can equate each factor to zero.
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  3. #3
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    Re: De Moivre's Theorem of Complex numbers Question Help

    Quote Originally Posted by BobP View Post
    The LHS will factorise. Then you can equate each factor to zero.
    And it factorises to...

    $\displaystyle \displaystyle \begin{align*} z^9 + z^5 - z^4 - 1 &\equiv z^9 - z^4 + z^5 - 1 \\ &\equiv z^4\left(z^5 - 1\right) + 1\left(z^5 - 1\right) \\ &\equiv \left(z^5 - 1\right)\left(z^4 + 1\right) \end{align*}$

    So that means you can set $\displaystyle \displaystyle \begin{align*} z^5 - 1 = 0 \end{align*}$ and $\displaystyle \displaystyle \begin{align*} z^4 + 1 = 0 \end{align*}$ and solve using DeMoivre's Theorem
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    Re: De Moivre's Theorem of Complex numbers Question Help

    Quote Originally Posted by Prove It View Post
    And it factorises to...

    $\displaystyle \displaystyle \begin{align*} z^9 + z^5 - z^4 - 1 &\equiv z^9 - z^4 + z^5 - 1 \\ &\equiv z^4\left(z^5 - 1\right) + 1\left(z^5 - 1\right) \\ &\equiv \left(z^5 - 1\right)\left(z^4 + 1\right) \end{align*}$

    So that means you can set $\displaystyle \displaystyle \begin{align*} z^5 - 1 = 0 \end{align*}$ and $\displaystyle \displaystyle \begin{align*} z^4 + 1 = 0 \end{align*}$ and solve using DeMoivre's Theorem
    Thanks alot!
    i am able to solve it now, it was pretty simple. thanks again
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