Problem 49 - Elementary and Intermediate Algebra

Question

Find the LCD of pair of rational expressions. \(\frac{3 x+1}{3 x-3}, \frac{3 x}{4 x-4}\)

Step-by-Step Solution

Answer

The LCD is \(12(x - 1)\).

Step 1: Identify the Denominators

The denominators of the given rational expressions are \(3x - 3\) and \(4x - 4\). Our first step will be to consider these expressions to find the Least Common Denominator (LCD).

Step 2: Factor Each Denominator

Factor each denominator: \(3x - 3 = 3(x - 1)\) and \(4x - 4 = 4(x - 1)\). Both denominators share a factor of \(x - 1\).

Step 3: Identify All Factors

Identify all distinct factors present in each denominator. We have the factors \(3\), \(4\), and \(x - 1\).

Step 4: Construct the LCD

The LCD is the product of all distinct factors in their highest powers. Thus, the LCD is \(12(x - 1)\) since \(12\) is the least common multiple of \(3\) and \(4\).

Key Concepts

Rational Expressions

Rational expressions are similar to fractions, but instead of integers, they involve polynomials. A rational expression is a fraction where the numerator and the denominator are polynomials, like \( \frac{3x+1}{3x-3} \) or \( \frac{3x}{4x-4} \). Rational expressions are a major topic in algebra because they allow us to work with polynomial functions in a fraction format, making it easier to analyze and manipulate them.

When you work with rational expressions, it is important to understand that:

• The denominator cannot be zero, since division by zero is undefined.
• They can often be simplified by factoring and cancelling common factors.
• Finding a Least Common Denominator (LCD) is essential when adding or subtracting them.

Understanding these basic properties helps you gain confidence in solving problems involving rational expressions, especially when combined with other algebraic concepts like factoring and finding least common multiples.

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that involves breaking down a polynomial into a product of its factors. This is particularly useful when working with rational expressions, as it simplifies the process of finding the Least Common Denominator.

Consider the polynomial expressions from our exercise: \(3x - 3\) and \(4x - 4\). Each of these can be factored as follows:

• \(3x - 3 = 3(x - 1)\)
• \(4x - 4 = 4(x - 1)\)

Notice that both expressions share a common factor of \(x - 1\). Factoring helps us by revealing these shared factors, which play a key role in finding the LCD.

When factoring polynomials, remember to:

• Look for a Greatest Common Factor (GCF) that can be factored out.
• Check if the polynomial can be factored further, using methods like grouping, difference of squares, or trinomial factoring.

Successfully factoring expressions is a foundational step in many algebra problems, making it easier to simplify and work with complex equations.

Least Common Multiple

The least common multiple (LCM) is the smallest multiple that is exactly divisible by each of the numbers in question. In the context of rational expressions, the LCM of the denominators is called the Least Common Denominator (LCD). The LCD is what we use to combine rational expressions through addition or subtraction.

In our exercise, the denominators after factoring are \(3(x - 1)\) and \(4(x - 1)\). Since both have \(x - 1\) as a factor, our task is to find the LCM of the constants \(3\) and \(4\). The steps are:

• List the multiples of each constant: \(3: 3, 6, 9, 12, \ldots\) and \(4: 4, 8, 12, \ldots\)
• Identify the smallest multiple that is common to both: \(12\)

Thus, the LCD is \(12(x - 1)\). Finding the LCD allows for the seamless operation of combining rational expressions, which is a typical requirement in algebra problems. Understanding the concept of LCM not only aids in algebra but also benefits arithmetic and number theory.