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Question Number 146083 by mathmax by abdo last updated on 10/Jul/21

$$\mathrm{F}\left(\mathrm{x}\right)={\mathrm{x}}^{\mathrm{n}}-{\mathrm{e}}^{\mathrm{in}\alpha }$$$$1)\mathrm{roots}\mathrm{of}\mathrm{F}\left(\mathrm{x}\right)?$$$$2)\mathrm{factorize}\mathrm{F}\left(\mathrm{x}\right)\mathrm{inside}\mathrm{C}\left[\mathrm{x}\right]$$

Answered by Olaf_Thorendsen last updated on 10/Jul/21

$$1)x^n=e^{in\alpha }$$$$x=e^{i(\alpha +\frac{2k\pi }{n})},k=0,1,...,k-1$$$$2)$$$$\mathrm{F}\left(x\right)=\underset{k=0}{\prod ^{n-1}}(x-e^{i(\alpha +\frac{2k\pi }{n})})$$

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