Problem 13 - Algebra and Trigonometry

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 the number to be excluded from the domain is \(x=6\).

Step 1: Factorize Numerator and Denominator

Factorize the expressions in the numerator and denominator. The numerator \(x^{2}+12x+36\) can be factored into \((x+6)(x+6)\), and the denominator \(x^{2}-36\) can be factored into \((x-6)(x+6)\).

Step 2: Simplify the Expression

After factorization, the expression becomes \(\frac{(x+6)(x+6)}{(x-6)(x+6)}\). By cancelling out the common factor of \((x+6)\), we get \(\frac{(x+6)}{(x-6)}\) as the simplified expression.

Step 3: Identify the Excluded Numbers

The denominator cannot be zero, so set \(x-6=0\) and solve for \(x\). This gives \(x=6\) as a value that must be excluded from the domain of the simplified expression.

Key Concepts

Domain of Rational Expressions

The domain of a rational expression includes all the possible values that you can substitute for the variable, without causing the denominator to be zero. In any rational expression, a zero in the denominator is undefined, which means we need to exclude such values from the domain. When determining the domain for the rational expression \( \frac{x^{2}+12x+36}{x^{2}-36} \), you'd set the denominator equal to zero and solve that equation to find the excluded values:

• First, identify the denominator: \( x^{2} - 36 \).
• Solve the equation \( x^{2} - 36 = 0 \).
• This equation simplifies to \( (x-6)(x+6) = 0 \), meaning \( x = 6 \) and \( x = -6 \) are the excluded values.

Therefore, the domain of the expression can include all real numbers except \( x = 6 \) and \( x = -6 \). Understanding domains helps avoid the undefined circumstances that arise when you divide by zero.

Factoring Quadratic Expressions

Factoring quadratic expressions plays a pivotal role in simplifying rational expressions like the one given: \( \frac{x^{2}+12x+36}{x^{2}-36} \). Factoring transforms complex expressions into easier terms that can be managed more efficiently. The steps for factoring usually involve identifying expressions that are perfect squares or trinomial quadratics:

• In the numerator, \( x^{2}+12x+36 \), notice it's a perfect square trinomial. As it can be represented as \( (x+6)(x+6) \).
• For the denominator, \( x^{2}-36 \), it is a difference of squares, expressed as \( (x-6)(x+6) \).

Once we've factored, we can easily identify and cancel any common factors in both the numerator and the denominator, which helps simplify the expression considerably to \( \frac{x+6}{x-6} \). Mastering the different techniques for factoring quadratics is essential for efficiently simplifying rational expressions.

Excluded Values in Algebra

In algebra, excluded values are numbers that make the denominator of a rational expression zero, rendering the expression undefined. It's crucial to identify and understand these values to prevent errors when dealing with rational expressions.For the expression \( \frac{(x+6)(x+6)}{(x-6)(x+6)} \), there's a little more to consider than just the obvious restrictions:

• Clearly, \( x-6 = 0 \) leads us to exclude \( x = 6 \) since it makes the denominator zero.
• However, observe that \( x+6 = 0 \) also impacts the original expression before simplification. While it cancels out, \( x = -6 \) should still not be a part of the domain, as it was a factor in the original denominator.

A proper grasp of excluded values helps avoid mistakes in algebraic simplifications and ensures that you understand the true domain of the expression. It's essential to account for both the fully simplified form and the original expression when analyzing excluded values.