Problem: Let S=(x−1)4+4(x−1)3+6(x−1)2+4(x−1)+1S=(x-1)^{4}+4(x-1)^{3}+6(x-1)^{2}+4(x-1)+1S=(x−1)4+4(x−1)3+6(x−1)2+4(x−1)+1. Then SSS equals:
Answer Choices:
A. (x−2)4(x-2)^{4}(x−2)4
B. (x−1)4(x-1)^{4}(x−1)4
C. x4x^{4}x4
D. (x+1)4(x+1)^{4}(x+1)4
E. x4+1x^{4}+1x4+1 Solution:
Consider the binomial expansion, (a+1)4(a+1)^{4}(a+1)4 =a4+4a3+6a2+4a+1=a^{4}+4 a^{3}+6 a^{2}+4 a+1=a4+4a3+6a2+4a+1. Let x−1=ax-1=ax−1=a. Then S=(x−1+1)4=x4S=(x-1+1)^{4}=x^{4}S=(x−1+1)4=x4.