Problem: Let g(x)=x5+x4+x3+x2+x+1. What is the remainder when the polynomial g(x12) is divided by the polynomial g(x) ?
Answer Choices:
A. 6
B. 5−x
C. 4−x+x2
D. 3−x+x2−x3
E. 2−x+x2−x3+x4
Solution:
Replacing x by x6 in the equation
(x−1)(xn+xn−1+…+1)=xn+1−1
yields
(x6−1)(x6n+x6(n−1)+…+1)=x6(n+1)−1
Thus
g(x12)g(x12)/g(x)=x60+…+x12+1=(x60−1)+…+(x12−1)+6=(x6−1)(x54+…)+…+(x6−1)(x6+1)+6=g(x)(x−1)[(x54+…)+…+(x6+1)]+6=(x−1)[(x54+…)+…+(x6−1)]+g(x)6,
and the remainder is 6.