Problem 33 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x+5}{x^{2}-25}$$

Step-by-Step Solution

Answer

The simplified form of the rational expression is \(\frac{1}{x-5}\) for \(x \neq -5\).

Step 1: Factorize the denominator

To simplify the given rational expression it's necessary to factorize the denominator. The expression \(x^{2}-25\) is a difference of two squares, which can be factored into \((x-5)(x+5)\).

Step 2: Simplify the expression

After the factorization of the denominator, the expression becomes \(\frac{x+5}{(x-5)(x+5)}\). Since \(x+5\) is common in the numerator and the denominator, those can be cancelled out. However, it's necessary to remember that \(x \neq -5\) because that would make the denominator equal to zero, which is undefined in the real number system.

Step 3: Write the simplified expression

After simplification, the resulting expression is \(\frac{1}{x-5}\).