Question Number 58612 by naka3546 last updated on 26/Apr/19
$${P}\left({x}\right)\:\:{is}\:\:{a}\:\:{monic}−{fifth}\:\:{degree}\:\:{polynomial}\:\:{that}\:{satisfy} \\ $$$$\:\:\:\:{P}\left(\mathrm{1}\right)\:\:=\:\:\mathrm{1} \\ $$$$\:\:\:\:{P}\left(\mathrm{2}\right)\:\:=\:\:\mathrm{4} \\ $$$$\:\:\:\:{P}\left(\mathrm{3}\right)\:\:=\:\:\mathrm{9} \\ $$$$\:\:\:\:{P}\left(\mathrm{4}\right)\:\:=\:\:\mathrm{16} \\ $$$$\:\:\:\:{P}\left(\mathrm{5}\right)\:\:=\:\:\mathrm{25} \\ $$$$\:\:\:\:{P}\left(\mathrm{6}\right)\:\:=\:\:? \\ $$$$ \\ $$
Answered by tanmay last updated on 26/Apr/19
$${p}\left({x}\right)=\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}−\mathrm{3}\right)\left({x}−\mathrm{4}\right)\left({x}−\mathrm{5}\right)+{x}^{\mathrm{2}} \\ $$$${p}\left(\mathrm{6}\right)=\mathrm{5}×\mathrm{4}×\mathrm{3}×\mathrm{2}×\mathrm{1}+\mathrm{6}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:=\mathrm{120}+\mathrm{36}=\mathrm{156} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$