Problem: If q1(x) and r1 are the quotient and remainder, respectively, when the polynomial x8 is divided by x+21, and if q2(x) and r2 are the quotient and remainder, respectively, when q1(x) is divided by x+21, then r2 equals
Answer Choices:
A. 2561
B. −161
C. 1
D. −16
E. 256
Solution:
Let a=−21. Applying the remainder theorem yields r1=a8, and solving the equality xs=(x−a)q1(x)+r1 for q1(x) yields
q1(x)=x−ax8−a8=x7+ax6+⋯+a7
[or, by factoring a difference of squares three times,
q1(x)=(x4+a4)(x2+a2)(x+a)]
Applying the remainder theorem to determine the remainder when q1(x) is divided by x−a yields
r2=q1(a)=8a7=−161