Problem 13 - College Algebra Essentials

Question

Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$\frac{x^{2}+12 x+36}{x^{2}-36}$$

Step-by-Step Solution

Answer

The simplified rational expression is \(\frac{x+6}{x-6}\). And, 'x' must not be equal to 6.

Step 1: Identify the given expression

The given rational expression is \(\frac{x^{2}+12 x+36}{x^{2}-36}\)

Step 2: Factor the numerator and the denominator

The numerator \(x^{2}+12 x+36\) can factor into \((x+6)^2\) and the denominator \(x^{2}-36\) can factor into \((x-6)(x+6)\). So, the rational expression becomes \(\frac{(x+6)^2}{(x-6)(x+6)}\).

Step 3: Simplify the expression

The \(x + 6\) terms in both the numerator and denominator can be canceled out, leading the expression to simplify as \(\frac{x+6}{x-6}\).

Step 4: Find the excluded numbers

The denominator of the rational expression (after simplification) cannot be zero. Therefore, we must solve \(x - 6 != 0\). Hence, \(x != 6\).