Problem 70 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}-4}{2-x}$$

Step-by-Step Solution

Answer

The simplified rational expression is \(-(x+2)\)

Step 1: Factor the numerator

The given rational expression is \(\frac{x^{2}-4}{2-x}\). The numerator is a difference of squares and can be factored as \((x-2)(x+2)\). So, we get \(\frac{(x-2)(x+2)}{2-x}\)

Step 2: Rewrite the denominator

Now, rewrite the denominator \(2-x\) as \(-(x-2)\). So, our rational expression becomes \(\frac{(x-2)(x+2)}{-(x-2)}\)

Step 3: Cancel the common factors

Now we see that \(x-2\) is a common factor in the numerator and the denominator. We can cancel these factors out. Thus our expression simplifies to \(\frac{-(x+2)}{1}\) which is \(-(x+2)\)