Roots of Unity: Level 5 Challenges Practice Problems Online

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

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

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

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

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

If $x^{2}-x+1=0$ , then find the value of $\left ( x-\frac{1}{x} \right )^{2}+\sum_{a=1}^{2015} \left ( x^a+\frac{1}{x^a} \right )^{2}$

If $w=e^{2\pi{i}/2015},$ find $\sum_{k=1}^{2014}\frac{1}{1+w^k+w^{2k}}.$

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 $s.$

A finite number of complex solutions none of which is real. Only one solution that can be real or complex. A finite number of complex solutions some of which are real. No complex or real solutions. Infinitely many complex solutions some of which are real. Infinitely many complex solutions none of which is real.

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Roots of Unity: Level 5 Challenges