Problem 69 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}-1}{1-x}$$

Step-by-Step Solution

Answer

The simplified form of the rational expression \(\frac{x^{2}-1}{1-x}\) is \(x+1\).

Step 1: Factorize the numerator

Start by factorizing the numerator. The expression \(x^{2}-1\) can be written as \((x-1)(x+1)\) as it's a difference of squares.

Step 2: Simplify the rational expression

Now the original expression is \(\frac{(x-1)(x+1)}{1-x}\). Notice here if we flip the signs in the denominator to \(x-1\), the original expression can be written as \(\frac{(x-1)(x+1)}{x-1}\)

Step 3: Cancel out common factors

We can cancel the common factor of \(x-1\) from the numerator and denominator. So the final simplified expression is \(x+1\)