Problem: If x−1−1x^{-1}-1x−1−1 is divided by x−1x-1x−1 the quotient is:
Answer Choices:
A. 111
B. 1x−1\dfrac{1}{x-1}x−11
C. −1x−1\dfrac{-1}{x-1}x−1−1
D. 1x\dfrac{1}{x}x1
E. −1x-\dfrac{1}{x}−x1 Solution:
x−1−1=1x−1=1−xx,1−xx⋅1x−1=−1xx^{-1}-1=\dfrac{1}{x}-1=\dfrac{1-x}{x}, \quad \dfrac{1-x}{x} \cdot \dfrac{1}{x-1}=-\dfrac{1}{x}x−1−1=x1−1=x1−x,x1−x⋅x−11=−x1.