Problem 47 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x^{2}+12 x+36}{x^{2}-36}$$

Step-by-Step Solution

Answer

The simplified form of the rational expression \(\frac{x^{2}+12 x+36}{x^{2}-36}\) is \(\frac{x + 6}{x - 6}\).

Step 1: Factorize the numerator

Start by factorizing the numerator. The expression \(x^{2} + 12x + 36\) can be written as \((x + 6)^{2}\).

Step 2: Factorize the denominator

Similarly, factorize the denominator. The expression \(x^{2} - 36\) can be factorized since it is a difference of squares. It can be written as \((x - 6)(x + 6)\) after factorization.

Step 3: Simplify the rational expression

The rational expression simplifies by cancelling common factors in the numerator and denominator. After cancellation, \(\frac{(x + 6)(x + 6)}{(x - 6)(x + 6)}\) simplifies to \(\frac{x + 6}{x - 6}\).