If x^-1-1 is divided by x-1 the quotient is:

1956 AMC 12 Problem 12

Attempt Problem 12 of the 1956 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1956 AMC 12 solutions, or check the answer key.

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12.

x^{-1}-1

is divided by

x-1

the quotient is:

1

\dfrac1{x-1}

-\dfrac1{x-1}

\dfrac1x

-\dfrac1x

Rewrite

x^{-1}-1

\frac{1}{x}-1

Factor

1-x=-(x-1)

Where the quotient is defined,

\begin{aligned} \frac{x^{-1}-1}{x-1} &=\frac{\frac{1-x}{x}}{x-1}\\ &=\frac{-(x-1)}{x(x-1)}\\ &=-\frac1x. \end{aligned}

Thus, the correct answer is E.