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Let \(\omega \)be a complex number such that \(\omega^3 = 1\). Find all possible values of \(\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}.\) Enter all the possible values, separated by commas.

 Apr 16, 2019

Best Answer 

 #1
avatar+22554 
+3

Let

\(\omega\)

be a complex number such that

\(\omega^3 = 1\).
Find all possible values of

\(\dfrac{1}{1 + \omega} + \dfrac{1}{1 + \omega^2}\).

 

\(\begin{array}{|rcll|} \hline && \mathbf{\dfrac{1}{1 + \omega} + \dfrac{1}{1 + \omega^2}} \\\\ &=& \dfrac{1 + \omega^2+1 + \omega}{(1 + \omega)(1 + \omega^2) } \\\\ &=& \dfrac{2 + \omega + \omega^2 }{1 + \omega^2 + \omega + \omega^3 } \quad | \quad \omega^3 = 1 \\\\ &=& \dfrac{2 + \omega + \omega^2 }{1 + \omega^2 + \omega + 1 } \\\\ &=& \dfrac{2 + \omega + \omega^2 }{2 + \omega + \omega^2} \\\\ &\mathbf{=}& \mathbf{1} \\ \hline \end{array}\)

 

laugh

 Apr 17, 2019
 #1
avatar+22554 
+3
Best Answer

Let

\(\omega\)

be a complex number such that

\(\omega^3 = 1\).
Find all possible values of

\(\dfrac{1}{1 + \omega} + \dfrac{1}{1 + \omega^2}\).

 

\(\begin{array}{|rcll|} \hline && \mathbf{\dfrac{1}{1 + \omega} + \dfrac{1}{1 + \omega^2}} \\\\ &=& \dfrac{1 + \omega^2+1 + \omega}{(1 + \omega)(1 + \omega^2) } \\\\ &=& \dfrac{2 + \omega + \omega^2 }{1 + \omega^2 + \omega + \omega^3 } \quad | \quad \omega^3 = 1 \\\\ &=& \dfrac{2 + \omega + \omega^2 }{1 + \omega^2 + \omega + 1 } \\\\ &=& \dfrac{2 + \omega + \omega^2 }{2 + \omega + \omega^2} \\\\ &\mathbf{=}& \mathbf{1} \\ \hline \end{array}\)

 

laugh

heureka Apr 17, 2019

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