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A root of unity is a complex number that, when raised to a positive integer power, results in 1. Roots of unity have applications to the geometry of regular polygons, group theory, and number theory.

\[\large \frac{31}{2- \alpha_1}+ \frac{31}{2- \alpha_2}+\frac{31}{2- \alpha_3} +\frac{31}{2- \alpha_4}\]

Given that \(1,\alpha_1, \alpha_2, \alpha_3,\alpha_4\) are distinct fifth roots of unity, evaluate the expression above.

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\[\large \sum _{ k=1 }^{ 20 }{ \left( { \alpha }^{ 3k }+{ \beta }^{ 3k }+{ \xi }^{ 3k } \right) }\]

\[ \begin{cases} {f\left( -1 \right) =-9} \\ {f\left( 1 \right) =-7 } \\ {f\left( 3 \right) =19} \end{cases} \]

If \(f\left( x \right) \) is monic cubic polynomial having roots \(\alpha ,\beta ,\xi \). Then evaluate topmost expression modulo 17.

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If \(w=e^{2\pi{i}/2015}\), find \[\sum_{k=1}^{2014}\frac{1}{1+w^k+w^{2k}}\]

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The equation \(x^3=-1\) has three solutions, one of which is real and the other two are non-real complex numbers. Determine the number and type of solutions of \[ \large x^{\frac{1}{\sqrt{2}}}=-1\]

**Note**: When \(x\) is a complex number different from \(0\), and \(r\) is a real number, \(x^r\) can have more than one possible value. In this case, we assume that the complex number \(x\) is a solution of the equation \(x^r=s,\) where \(s\) is a given real number, if at least one of the values of \(x^r\) is equal to \(s.\)

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