+1245 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.
+26393 Let
\(\omega\)
be a complex number such that
\(\omega^3 = 1\).
Find all possible values of
\(\dfrac{1}{1 + \omega} + \dfrac{1}{1 + \omega^2}\).
\(\begin{array}{|rcll|} \hline && \mathbf{\dfrac{1}{1 + \omega} + \dfrac{1}{1 + \omega^2}} \\\\ &=& \dfrac{1 + \omega^2+1 + \omega}{(1 + \omega)(1 + \omega^2) } \\\\ &=& \dfrac{2 + \omega + \omega^2 }{1 + \omega^2 + \omega + \omega^3 } \quad | \quad \omega^3 = 1 \\\\ &=& \dfrac{2 + \omega + \omega^2 }{1 + \omega^2 + \omega + 1 } \\\\ &=& \dfrac{2 + \omega + \omega^2 }{2 + \omega + \omega^2} \\\\ &\mathbf{=}& \mathbf{1} \\ \hline \end{array}\)
+26393 Let
\(\omega\)
be a complex number such that
\(\omega^3 = 1\).
Find all possible values of
\(\dfrac{1}{1 + \omega} + \dfrac{1}{1 + \omega^2}\).
\(\begin{array}{|rcll|} \hline && \mathbf{\dfrac{1}{1 + \omega} + \dfrac{1}{1 + \omega^2}} \\\\ &=& \dfrac{1 + \omega^2+1 + \omega}{(1 + \omega)(1 + \omega^2) } \\\\ &=& \dfrac{2 + \omega + \omega^2 }{1 + \omega^2 + \omega + \omega^3 } \quad | \quad \omega^3 = 1 \\\\ &=& \dfrac{2 + \omega + \omega^2 }{1 + \omega^2 + \omega + 1 } \\\\ &=& \dfrac{2 + \omega + \omega^2 }{2 + \omega + \omega^2} \\\\ &\mathbf{=}& \mathbf{1} \\ \hline \end{array}\)
