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​