Question Number 113928 by ZiYangLee last updated on 16/Sep/20

Find the remainder when x^(2006) −1   is divided by x^4 +x^3 +2x^2 +x+1.

Answered by MJS_new last updated on 16/Sep/20

x^4 +x^3 +2x^2 +x+1=(x^2 +1)(x^2 +x+1)  the remaining fractions of  ((x^(2n) −1)/((x^2 +1)(x^2 +x+1)))  are  0 for n=6k  ((x^2 −1)/((x^2 +1)(x^2 +x+1))) for n=6k+1 [2006=2(6×167+1)]  −((x+2)/(x^2 +x+1)) for n=6k+2  ((2x)/(x^2 +1)) for n=6k+3  −((2x+1)/(x^2 +x+1)) for n=6k+4  ((x^3 +x−2)/((x^2 +1)(x^2 +x+1))) for n=6k+5

Commented by1549442205PVT last updated on 16/Sep/20

For x^(2006) −1⇒n=1003=6×167+1

Commented byMJS_new last updated on 16/Sep/20

you are right, thank you