Question

which shows the expression \\(\frac{x^2 - 1}{x^2 - x}\\) in simplest form?\
\\(\frac{1}{x}\\)\
\\(\frac{x + 1}{x}\\)\
\\(\frac{x - 1}{x}\\)\
\\(\frac{1}{x + 1}\\)

Explanation:

Step1: Factor numerator and denominator

Factor \(x^2 - 1\) as a difference of squares: \(x^2 - 1=(x + 1)(x - 1)\).
Factor \(x^2 - x\) by taking out the common factor \(x\): \(x^2 - x=x(x - 1)\).
So the expression becomes \(\frac{(x + 1)(x - 1)}{x(x - 1)}\).

Step2: Cancel common factors

The common factor \((x - 1)\) (where \(x
eq1\) to avoid division by zero) can be canceled from the numerator and the denominator.
After canceling, we get \(\frac{x + 1}{x}\).

Answer:

\(\boldsymbol{\frac{x + 1}{x}}\) (corresponding to the option with \(\frac{x + 1}{x}\))