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?
Answered by EDWIN88 last updated on 01/Apr/21
$$\mathrm{Apply}\:\mathrm{the}\:\mathrm{Factor}\:\mathrm{theorem}\:.\:\mathrm{That}\:\mathrm{means} \\ $$$$\mathrm{to}\:\mathrm{write}\:{x}^{\mathrm{2}} \:=\:−{x}−\mathrm{1}\:\mathrm{and}\:\mathrm{observe}\:\mathrm{that} \\ $$$${x}^{\mathrm{3}} \:=−{x}^{\mathrm{2}} −{x}={x}+\mathrm{1}−{x}=\mathrm{1} \\ $$$${x}^{\mathrm{4}} \:=\:{x}\:;\:{x}^{\mathrm{5}} \:=\:−{x}−\mathrm{1}\:,\:{x}^{\mathrm{6}} =\:\mathrm{1},\:{x}^{\mathrm{7}} ={x} \\ $$$$\mathrm{it}\:\mathrm{follows}\:\mathrm{that}\: \\ $$$$\mathrm{3}{x}^{\mathrm{75}} +\mathrm{2}{x}^{\mathrm{14}} −\mathrm{3}{x}^{\mathrm{2}} −\mathrm{3}\:=\:\mathrm{3}\left(\mathrm{1}\right)+\mathrm{2}\left({x}^{\mathrm{2}} \right)−\mathrm{3}{x}^{\mathrm{2}} −\mathrm{1} \\ $$$$=\:\mathrm{2}−{x}^{\mathrm{2}} \:=\mathrm{2}−\left(−{x}−\mathrm{1}\right)=\:{x}+\mathrm{3} \\ $$