Question

Simplify the rational expression.

$\frac{x^2-x-2}{x^2-1}$

Short answer

The simplified rational expression is$\frac{x-2}{x-1}$

Step-by-step solution

Step 1. Apply the concept of simplifying rational expressions.

To simplify rational expressions, we factor both numerator and denominator and use the following property of fractions:

$\frac{AC}{BC}=\frac{A}{B}$

This allows us to cancel common factors from the numerator and denominator.

Step 2. Analyze the expression.

Given$\frac{x^2-x-2}{x^2-1}$

This is a rational expression.

To simplify the rational expression, we factor both numerator and denominator and use this$\frac{AC}{BC}=\frac{A}{B}$

Therefore, to simplify a rational expression, we cancel factors that are common to the numerator and denominator.

Step 3. Simplify the expression.

Consider

$\begin{array}{l}\frac{x^2-x-2}{x^2-1}=\frac{x^2-2x+x-2}{\left(x+1\right)\left(x-1\right)}\mathrm{............}\left[a^2-b^2=\left(a-b\right)\left(a+b\right)\right]\\ \Rightarrow \frac{x^2-x-2}{x^2-1}=\frac{x\left(x-2\right)+\left(x-2\right)}{\left(x+1\right)\left(x-1\right)}\mathrm{.....}\left[a\left(b+c\right)=ab+ac\right]\\ \Rightarrow \frac{x^2-x-2}{x^2-1}=\frac{\left(x-2\right)\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}\mathrm{.....}\left[a\left(b+c\right)=ab+ac\right]\end{array}$

The common factor to the numerator and denominator is$x+1$

So we will cancel the common factor from the numerator and denominator.

The simplified expression is$\frac{\left(x-2\right)\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}=\frac{x-2}{x-1}$