Question Number 154931 by qaz last updated on 23/Sep/21
$$\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{2x}^{\mathrm{5}} +\mathrm{2x}^{\mathrm{4}} −\mathrm{53x}^{\mathrm{3}} −\mathrm{57x}+\mathrm{54}\right)^{\mathrm{2011}} \\ $$$$\mathrm{f}\left(\frac{\sqrt{\mathrm{111}}−\mathrm{1}}{\mathrm{2}}\right)=? \\ $$
Answered by MJS_new last updated on 23/Sep/21
$$−\mathrm{1} \\ $$
Commented by qaz last updated on 23/Sep/21
$$\mathrm{I}\:\mathrm{have}\:\mathrm{no}\:\mathrm{answer} \\ $$
Answered by mindispower last updated on 23/Sep/21
$${x}=\:\frac{\sqrt{\mathrm{111}}−\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{111} \\ $$$$\mathrm{4}{x}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{100}=\mathrm{0} \\ $$$${x}^{\mathrm{2}} +{x}=\mathrm{25},{x}^{\mathrm{2}} =\mathrm{25}−{x} \\ $$$${x}^{\mathrm{3}} =\mathrm{25}{x}−{x}^{\mathrm{2}} =\mathrm{26}{x}−\mathrm{25} \\ $$$${x}^{\mathrm{4}} =\mathrm{26}{x}^{\mathrm{2}} −\mathrm{25}{x}=\mathrm{26}\left(\mathrm{25}−{x}\right)−\mathrm{25}{x} \\ $$$$=−\mathrm{51}{x}+\mathrm{650} \\ $$$${x}^{\mathrm{5}} =−\mathrm{51}{x}^{\mathrm{2}} +\mathrm{650}{x}...... \\ $$$$ \\ $$$$ \\ $$