Question Number 201857 by cortano12 last updated on 14/Dec/23
$$\:\:\mathrm{If}\:{a},{b},{c},{d},{e}\:\mathrm{are}\:\mathrm{thr}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$$\:\:\mathrm{2x}^{\mathrm{5}} −\mathrm{3x}^{\mathrm{3}} +\mathrm{2x}−\mathrm{7}=\mathrm{0}\:,\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\:\mathrm{value}\:\mathrm{of}\:\underset{\mathrm{cyc}} {\prod}\left({a}^{\mathrm{3}} −\mathrm{1}\right)\: \\ $$
Commented by Frix last updated on 14/Dec/23
$$\mathrm{I}\:\mathrm{found}\:\mathrm{108} \\ $$
Answered by Frix last updated on 14/Dec/23
![f(x)=x^5 +bx^3 +dx+e=0 Π_(cyc) (x^3 −1)=−f(1)f(−(1/2)+((√3)/2)i)f(−(1/2)−((√3)/2)i)= =(b+e)^3 +d^3 −3d(b+e)+1 [Also works for f(x)=x^5 +ax^4 +bx^3 +cx^2 +dx+e]](https://www.tinkutara.com/question/Q201875.png)
$${f}\left({x}\right)={x}^{\mathrm{5}} +{bx}^{\mathrm{3}} +{dx}+{e}=\mathrm{0} \\ $$$$\underset{\mathrm{cyc}} {\prod}\left({x}^{\mathrm{3}} −\mathrm{1}\right)=−{f}\left(\mathrm{1}\right){f}\left(−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{i}\right){f}\left(−\frac{\mathrm{1}}{\mathrm{2}}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{i}\right)= \\ $$$$=\left({b}+{e}\right)^{\mathrm{3}} +{d}^{\mathrm{3}} −\mathrm{3}{d}\left({b}+{e}\right)+\mathrm{1} \\ $$$$\left[\mathrm{Also}\:\mathrm{works}\:\mathrm{for}\:{f}\left({x}\right)={x}^{\mathrm{5}} +{ax}^{\mathrm{4}} +{bx}^{\mathrm{3}} +{cx}^{\mathrm{2}} +{dx}+{e}\right] \\ $$
Commented by Frix last updated on 14/Dec/23
$$\mathrm{Works}\:\mathrm{for}\:\mathrm{any}\:\mathrm{polynome} \\ $$$$\underset{\mathrm{cyc}} {\prod}\left({x}^{{m}} −\mathrm{1}\right)=−\underset{{n}=\mathrm{0}} {\overset{{m}−\mathrm{1}} {\prod}}{f}\left(\mathrm{e}^{\mathrm{i}\frac{\mathrm{2}\pi{n}}{{m}}} \right) \\ $$