Problem 69 - College Algebra with Modeling and Visualization

Question

Find the \(L C D\) for the rational expressions. $$ \frac{1}{x+4}, \frac{1}{x^{2}-16} $$

Step-by-Step Solution

Answer

The LCD is \((x+4)(x-4)\).

Step 1: Factor Each Denominator

First, examine the denominators of the given rational expressions: - The first denominator is \(x+4\), which cannot be factored further. - The second denominator is \(x^2 - 16\), which is a difference of squares and can be factored as \((x+4)(x-4)\).

Step 2: Identify All Unique Factors

Identify all unique factors from the denominators. From the first expression, we have \(x+4\). From the second expression, we have \((x+4)(x-4)\). Thus, the unique factors are \(x+4\) and \(x-4\).

Step 3: Determine the Least Common Denominator (LCD)

The least common denominator (LCD) is the product of all unique factors identified, each taken to the highest power that appears in any denominator. In this case, \((x+4)\) and \((x-4)\) each appear only once in their respective factorizations. Thus, the LCD is \((x+4)(x-4)\).

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are similar to numerical fractions but involve variables, which can make them more complex. The primary operations on rational expressions include addition, subtraction, multiplication, and division.

When dealing with these expressions, finding a common denominator is critical, especially for addition and subtraction, as it allows us to combine expressions into a single fraction. This process requires factoring the denominators to find the least common denominator (LCD).

In the given exercise, our rational expressions are \(\frac{1}{x+4}\) and \(\frac{1}{x^2-16}\). To find the LCD, we need to factor these denominators and identify all unique factors.

Factoring Denominators

Factoring denominators is an essential step in working with rational expressions. It involves breaking down a polynomial into simpler, multiplicative components, which can help in finding common denominators or simplifying expressions.

In the exercise, two types of factored expressions were considered:

• \(x+4\): This is already in its simplest form, as it cannot be factored further.
• \(x^2-16\): This is a more complex expression that can be simplified using specific factoring techniques, namely the "difference of squares" method.
Factoring allows us to clearly see the factors involved and understand how they contribute to the overall expression.

Difference of Squares

The difference of squares is a special case when factoring polynomials, occurring when you have an expression of the form \(a^2 - b^2\). The rule for factoring a difference of squares is given by: \[a^2 - b^2 = (a - b)(a + b)\].

In the exercise, \(x^2 - 16\) is identified as a difference of squares because it can be rewritten as \(x^2 - 4^2\). Applying the rule, it can be factored into \((x - 4)(x + 4)\).

Recognizing this pattern can simplify many problems involving polynomials and is particularly helpful in finding the least common denominator. This method allows us to express \(x^2 - 16\) in terms of its factors, identifying its contributions to the larger expression and ensuring we calculate an accurate least common denominator.