When `x^(-1)-1` is divided by `x-1`,the quotient is

Options

A. `-1`
B. `-(1)/(x)`
C. `(1)/(x^(2))`
D. `(1)/((x-1)^(2))`

Answer

To solve the problem of dividing \( x^{-1} - 1 \) by \( x - 1 \), let's go through the steps systematically.

### Step 1: Rewrite \( x^{-1} - 1 \)
We know that \( x^{-1} \) is the same as \( \frac{1}{x} \). Thus, we can rewrite the expression:
\[ x^{-1} - 1 = \frac{1}{x} - 1 \]
Now, we need a common denominator to combine these terms:
\[ \frac{1}{x} - 1 = \frac{1}{x} - \frac{x}{x} = \frac{1 - x}{x} \]

### Step 2: Set up the division
Now we need to divide \( \frac{1 - x}{x} \) by \( x - 1 \):
\[ \frac{\frac{1 - x}{x}}{x - 1} \]

### Step 3: Simplify the division
Dividing by \( x - 1 \) is the same as multiplying by its reciprocal:
\[ \frac{1 - x}{x} \cdot \frac{1}{x - 1} \]
Since \( 1 - x \) can be rewritten as \( - (x - 1) \):
\[ \frac{- (x - 1)}{x} \cdot \frac{1}{x - 1} \]

### Step 4: Cancel out the common terms
Now we can cancel \( (x - 1) \) in the numerator and denominator:
\[ \frac{-1}{x} \]

### Conclusion
Thus, the quotient when \( x^{-1} - 1 \) is divided by \( x - 1 \) is:
\[ -\frac{1}{x} \]

### Final Answer
The correct answer is \( -\frac{1}{x} \).

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