Question Number 137335 by liberty last updated on 01/Apr/21
$$ \\ $$ P(x) = 3x^75 + 2x^14 - 3x^2 - 1. What is the remainder when the above polynomial of s divided by x^2+x+1?\\n
Answered by EDWIN88 last updated on 01/Apr/21
$$\mathrm{Apply}\:\mathrm{the}\:\mathrm{Factor}\:\mathrm{theorem}\:.\:\mathrm{That}\:\mathrm{means} \\ $$ $$\mathrm{to}\:\mathrm{write}\:{x}^{\mathrm{2}} \:=\:−{x}−\mathrm{1}\:\mathrm{and}\:\mathrm{observe}\:\mathrm{that} \\ $$ $${x}^{\mathrm{3}} \:=−{x}^{\mathrm{2}} −{x}={x}+\mathrm{1}−{x}=\mathrm{1} \\ $$ $${x}^{\mathrm{4}} \:=\:{x}\:;\:{x}^{\mathrm{5}} \:=\:−{x}−\mathrm{1}\:,\:{x}^{\mathrm{6}} =\:\mathrm{1}\color{mathred}{,}\color{mathred}{\:}{x}^{\mathrm{7}} ={x} \\ $$ $$\mathrm{it}\:\mathrm{follows}\:\mathrm{that}\: \\ $$ $$\mathrm{3}{x}^{\mathrm{75}} +\mathrm{2}{x}^{\mathrm{14}} −\mathrm{3}{x}^{\mathrm{2}} −\mathrm{3}\:\color{mathred}{=}\color{mathred}{\:}\mathrm{\color{mathred}{3}}\color{mathred}{\left(}\mathrm{\color{mathred}{1}}\color{mathred}{\right)}\color{mathred}{+}\mathrm{\color{mathred}{2}}\color{mathred}{\left(}{\color{mathred}{x}}^{\mathrm{\color{mathred}{2}}} \color{mathred}{\right)}\color{mathred}{−}\mathrm{\color{mathred}{3}}{\color{mathred}{x}}^{\mathrm{\color{mathred}{2}}} \color{mathred}{−}\mathrm{\color{mathred}{1}} \\ $$ $$=\:\mathrm{\color{mathred}{2}}\color{mathred}{−}{\color{mathred}{x}}^{\mathrm{\color{mathred}{2}}} \color{mathred}{\:}\color{mathred}{=}\mathrm{\color{mathred}{2}}\color{mathred}{−}\color{mathred}{\left(}\color{mathred}{−}{\color{mathred}{x}}\color{mathred}{−}\mathrm{\color{mathred}{1}}\color{mathred}{\right)}\color{mathred}{=}\color{mathred}{\:}{\color{mathred}{x}}\color{mathred}{+}\mathrm{\color{mathred}{3}} \\ $$