Problem 13 - College Algebra

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 expression is \(\frac{(x+6)}{(x-6)}\). And we must exclude \(x=6\) from the domain as it makes the denominator equal to zero.

Step 1: Identify and Factorize the Quadratic Terms

We have \(x^{2}+12 x+36\) in the numerator. This is a quadratic expression in the form \(x^2 + 2ax + a^2\) which can be factored as \((x + a)^2\). Here, \(a = 6\), therefore \(x^{2}+12 x+36\) can be written as \((x+6)^2\). Similarly, the denominator \(x^{2}-36\) is a difference of squares which can be factored as \((x-6)(x+6)\).

Step 2: Simplify the Rational Expression

Now our expression is \(\frac{(x+6)^2}{(x-6)(x+6)}\). The term \((x+6)\) in the numerator and denominator can be cancelled out. Hence, our simplified expression is \(\frac{(x+6)}{(x-6)}\).

Step 3: Identify Values to Exclude from Domain

For a rational expression, we exclude values that make the denominator equal to zero as division by zero is undefined. Therefore, we find the value for x which makes the denominator zero. In this case, \(x-6=0\), hence, \(x=6\) is excluded from the domain of the simplified expression.