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Question Number 15180 by arnabpapu550@gmail.com last updated on 08/Jun/17

$$\mathrm{If}\omega \mathrm{is}\mathrm{a}\mathrm{cube}\mathrm{root}\mathrm{of}\mathrm{unity},\mathrm{then}\mathrm{a}\mathrm{root}$$$$\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{polynomial}\mathrm{is}$$$$\left|\begin{array}{ccc}x+1& \omega & {\omega }^2\\ \omega & x+{\omega }^2& 1\\ {\omega }^2& 1& x+\omega \end{array}\right|$$

Answered by ajfour last updated on 08/Jun/17

$$(x+1)(x^2-x)-{\omega }^{2}x+{\omega }^2(\omega -{\omega }^{2}x-1)=0$$$$x(x+1)(x-1)+x=0$$$$\Rightarrow x(x^2-1+1)=0$$$$or{x}^3=0\Rightarrow \mathit{x}=0.$$

Commented by arnabpapu550@gmail.com last updated on 08/Jun/17

$$\mathrm{Thank}\mathrm{you}\mathrm{sir}.$$

Commented by arnabpapu550@gmail.com last updated on 08/Jun/17

$$\mathrm{Please}\mathrm{solve}15191\mathrm{no}.\mathrm{question}\mathrm{too}.$$

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