Question Number 228335 by fantastic2 last updated on 07/Apr/26
$${x}+\frac{\mathrm{1}}{{x}}=\sqrt{\mathrm{3}} \\ $$$${x}^{\mathrm{30}} +{x}^{\mathrm{24}} +{x}^{\mathrm{18}} +{x}^{\mathrm{12}} +{x}^{\mathrm{6}} +\mathrm{1}=? \\ $$
Answered by AgniMath last updated on 07/Apr/26
$${x}+\frac{\mathrm{1}}{{x}}=\sqrt{\mathrm{3}} \\ $$$$\Rightarrow\:{x}^{\mathrm{3}} +\frac{\mathrm{1}}{{x}^{\mathrm{3}} }+\mathrm{3}\left({x}+\frac{\mathrm{1}}{{x}}\right)=\mathrm{3}\sqrt{\mathrm{3}} \\ $$$$\Rightarrow\:{x}^{\mathrm{3}} +\frac{\mathrm{1}}{{x}^{\mathrm{3}} }+\mathrm{3}\sqrt{\mathrm{3}}=\mathrm{3}\sqrt{\mathrm{3}} \\ $$$$\Rightarrow\:\frac{{x}^{\mathrm{6}} +\mathrm{1}}{{x}^{\mathrm{3}} }=\mathrm{0} \\ $$$$\Rightarrow\:{x}^{\mathrm{6}} +\mathrm{1}=\mathrm{0} \\ $$$$ \\ $$$${x}^{\mathrm{30}} +{x}^{\mathrm{24}} +{x}^{\mathrm{18}} +{x}^{\mathrm{12}} +{x}^{\mathrm{6}} +\mathrm{1} \\ $$$$=\:{x}^{\mathrm{24}} \left({x}^{\mathrm{6}} +\mathrm{1}\right)+{x}^{\mathrm{12}} \left({x}^{\mathrm{6}} +\mathrm{1}\right)+\left({x}^{\mathrm{6}} +\mathrm{1}\right) \\ $$$$=\:\mathrm{0} \\ $$
Commented by fantastic2 last updated on 07/Apr/26
$${great}! \\ $$