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# Help!

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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

#1
+25532
+3

Let

$$\omega$$

be a complex number such that

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

$$\dfrac{1}{1 + \omega} + \dfrac{1}{1 + \omega^2}$$.

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

Apr 17, 2019

#1
+25532
+3

Let

$$\omega$$

be a complex number such that

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

$$\dfrac{1}{1 + \omega} + \dfrac{1}{1 + \omega^2}$$.

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

heureka Apr 17, 2019