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Question Number 201857 by cortano12 last updated on 14/Dec/23

$$\mathrm{If}a,b,c,d,e\mathrm{are}\mathrm{thr}\mathrm{roots}\mathrm{of}$$$${2\mathrm{x}}^5-{3\mathrm{x}}^3+2\mathrm{x}-7=0,\mathrm{find}\mathrm{the}$$$$\mathrm{value}\mathrm{of}\prod _{\mathrm{cyc}}(a^3-1)$$

Commented by Frix last updated on 14/Dec/23

$$\mathrm{I}\mathrm{found}108$$

Answered by Frix last updated on 14/Dec/23

$$f\left(x\right)=x^5+{bx}^3+dx+e=0$$$$\prod _{\mathrm{cyc}}(x^3-1)=-f\left(1\right)f(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i})f(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i})=$$$$={(b+e)}^3+d^3-3d(b+e)+1$$$$[\mathrm{Also}\mathrm{works}\mathrm{for}f\left(x\right)=x^5+{ax}^4+{bx}^3+{cx}^2+dx+e]$$

Commented by Frix last updated on 14/Dec/23

$$\mathrm{Works}\mathrm{for}\mathrm{any}\mathrm{polynome}$$$$\prod _{\mathrm{cyc}}(x^m-1)=-\underset{n=0}{\prod ^{m-1}}f\left({\mathrm{e}}^{\mathrm{i}\frac{2\pi n}{m}}\right)$$

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