Problem 5 - Elementary and Intermediate Algebra

Question

What is the LCD for \(\frac{x-1}{x+6}\) and \(\frac{1}{x+3} ?\)

Step-by-Step Solution

Answer

LCD = \((x+6)(x+3)\)

Step 1: Identify the Denominators

The given rational expressions are \( \frac{x-1}{x+6} \) and \( \frac{1}{x+3} \). Here, the denominators are \( x+6 \) and \( x+3 \).

Step 2: Factor the Denominators

Since \( x+6 \) and \( x+3 \) are already linear polynomials, they do not have factors other than themselves. Therefore, the factored form of these denominators remains as \( x+6 \) and \( x+3 \).

Step 3: Determine the Least Common Denominator (LCD)

To find the LCD, identify the least common multiple of the denominators. Since \( x+6 \) and \( x+3 \) are distinct and cannot be further factored, the LCD is simply the product of these two terms: \((x+6)(x+3)\).

Key Concepts

Rational Expressions

Rational expressions are mathematical phrases that represent ratios of two polynomials. Just like fractions, they have a numerator and a denominator. When dealing with rational expressions, it is crucial to understand the structure and behavior of both the numerator and the denominator. In the exercise, we have two rational expressions:

• \( \frac{x-1}{x+6} \)
• \( \frac{1}{x+3} \)

The task is to find their least common denominator (LCD) so that they can be combined or compared easily. Understanding the components of these expressions is the first step. The numerators here are \(x-1\) and \(1\), and the denominators are \(x+6\) and \(x+3\) respectively. This sets the stage for finding the LCD, which requires analyzing these denominators closely.

Factoring Polynomials

Factoring polynomials is a key skill in working with rational expressions. It involves breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. In the context of the given exercise, we look at the denominators:

• \(x+6\)
• \(x+3\)

These are already linear polynomials, which means they are in their simplest form, and further factoring is not necessary. Linear polynomials like these cannot be factorized except into themselves.
Factoring is more relevant when dealing with higher degree polynomials (like quadratics), where terms need to be split into products of simpler terms to find common denominators or solve equations. So, for this exercise, factoring is straightforward, but in more complex scenarios, it involves looking for common factors or using techniques like grouping or the quadratic formula.

Least Common Multiple

The Least Common Multiple (LCM) is fundamental in finding the Least Common Denominator (LCD) of rational expressions. The LCM of the denominators provides a common ground for operations like addition, subtraction, and comparison.
For our expressions, the denominators are \(x+6\) and \(x+3\). To find the LCM here, since they are distinct and cannot be broken down further, we simply multiply them to find the LCD.
The product \[(x+6)(x+3)\]becomes the least common denominator. Finding the LCD involves evaluating either holding terms constant or finding shared factors among polynomials. In this case, as no shared factors exist and both terms are linear, the direct multiplication provides the simplest and correct LCM. This step ensures both expressions can be expressed over a common denominator, facilitating further mathematical operations.