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Chapter 0: Problem 13

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

Short Answer

Expert verified
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 by step solution

01

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 \((x-6)(x+6)\).
02

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 \(\frac{(x+6)}{(x-6)}\).
03

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.

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