Finding The Least Common Denominator LCD For Rational Expressions A Step By Step Guide

by ADMIN 87 views

Finding the least common denominator (LCD) is a crucial skill when working with rational expressions, especially when you need to add or subtract them. The LCD allows you to combine fractions with different denominators into a single fraction, simplifying the overall expression. In this article, we'll delve into how to find the LCD for the given rational expressions:

18r−4⋅−5rr2−5r+4⋅r+7r2−4r−5\frac{18}{r-4} \cdot \frac{-5r}{r^2-5r+4} \cdot \frac{r+7}{r^2-4r-5}

We will break down the process step by step, ensuring you understand each stage and can confidently tackle similar problems in the future. The correct answer among the choices provided is A. (r-1)(r-4)(r-5). Let's explore why this is the case.

Understanding the Least Common Denominator (LCD)

Before we dive into the specific problem, it’s essential to understand what the least common denominator actually is. The LCD is the smallest multiple that is common to a set of denominators. In simpler terms, it’s the smallest expression that each denominator can divide into evenly. For numerical fractions, this might be straightforward, but with rational expressions containing variables, we need to factorize the denominators first.

The least common denominator serves as a common ground for the fractions, allowing us to perform operations like addition and subtraction. Imagine trying to add fractions like 12\frac{1}{2} and 13\frac{1}{3}. You can't directly add them because they have different denominators. The LCD in this case is 6, which is the smallest number both 2 and 3 divide into. We rewrite the fractions as 36\frac{3}{6} and 26\frac{2}{6}, and then easily add them to get 56\frac{5}{6}. Similarly, with rational expressions, the LCD allows us to rewrite each fraction with a common denominator, making addition and subtraction possible.

In the context of rational expressions, finding the LCD involves identifying all unique factors in the denominators and taking the highest power of each factor. This ensures that the LCD is divisible by each denominator. This process often involves factoring quadratic or other polynomial expressions into simpler linear factors. By carefully factoring and identifying common and unique factors, we can construct the LCD that serves as the foundation for further operations with rational expressions.

Step 1: Factor Each Denominator

The first crucial step in finding the least common denominator is to factor each denominator completely. Factoring allows us to identify the individual components that make up each denominator, which is essential for determining the common and unique factors needed for the LCD. Let's break down the denominators from our given expression:

18r−4⋅−5rr2−5r+4⋅r+7r2−4r−5\frac{18}{r-4} \cdot \frac{-5r}{r^2-5r+4} \cdot \frac{r+7}{r^2-4r-5}

  1. The first denominator is (r - 4). This is already in its simplest form and cannot be factored further. It’s a linear expression, meaning it’s a polynomial of degree one. We simply note this factor and move on to the next denominator.
  2. The second denominator is (r² - 5r + 4). This is a quadratic expression, which means it’s a polynomial of degree two. We need to factor this into two binomials. We are looking for two numbers that multiply to 4 and add up to -5. These numbers are -1 and -4. Therefore, we can factor the quadratic as (r - 1)(r - 4).
  3. The third denominator is (r² - 4r - 5). This is another quadratic expression. We need to find two numbers that multiply to -5 and add up to -4. These numbers are -5 and 1. Therefore, we can factor the quadratic as (r - 5)(r + 1).

Now that we have factored each denominator, we have the following expressions:

18(r−4)⋅−5r(r−1)(r−4)⋅r+7(r−5)(r+1)\frac{18}{(r-4)} \cdot \frac{-5r}{(r-1)(r-4)} \cdot \frac{r+7}{(r-5)(r+1)}

Factoring the denominators is a critical step. It transforms complex expressions into products of simpler terms, making it easier to identify common and unique factors. This simplification is the cornerstone of finding the LCD and is indispensable for performing operations with rational expressions.

Step 2: Identify All Unique Factors

After successfully factoring the denominators, the next step in finding the least common denominator is to identify all unique factors present in the expressions. Unique factors are the distinct terms that appear in any of the denominators. These factors will form the building blocks of our LCD. Let's revisit our factored expressions:

18(r−4)⋅−5r(r−1)(r−4)⋅r+7(r−5)(r+1)\frac{18}{(r-4)} \cdot \frac{-5r}{(r-1)(r-4)} \cdot \frac{r+7}{(r-5)(r+1)}

Now, let’s list out all the unique factors:

  1. (r - 4): This factor appears in the first and second denominators.
  2. (r - 1): This factor is present in the second denominator.
  3. (r - 5): This factor is found in the third denominator.
  4. (r + 1): This factor also appears in the third denominator.

Each of these factors is unique and must be included in the LCD to ensure that the LCD is divisible by each original denominator. We don't need to list a factor more than once, even if it appears multiple times within a single denominator (as (r - 4) does). Identifying these unique factors is crucial because the LCD must contain each of them to properly serve as a common denominator.

The process of identifying unique factors is akin to gathering the necessary ingredients for a recipe. Each factor plays a critical role in the final LCD, just as each ingredient is essential for the dish. By meticulously listing each unique factor, we ensure that our LCD will be comprehensive and correct, setting the stage for further operations with rational expressions.

Step 3: Determine the Highest Power of Each Factor

Once we have identified all the unique factors, the subsequent step in finding the least common denominator involves determining the highest power of each unique factor present in any of the denominators. This is crucial because the LCD must be divisible by each denominator, and thus, it must include each factor raised to its highest power. Let’s consider our expressions again:

18(r−4)⋅−5r(r−1)(r−4)⋅r+7(r−5)(r+1)\frac{18}{(r-4)} \cdot \frac{-5r}{(r-1)(r-4)} \cdot \frac{r+7}{(r-5)(r+1)}

We've already identified the unique factors as: (r - 4), (r - 1), (r - 5), and (r + 1). Now we need to determine the highest power of each factor:

  1. (r - 4): In the first denominator, it appears as (r - 4)¹; in the second denominator, it appears as (r - 4)¹; it does not appear in the third denominator. The highest power is 1.
  2. (r - 1): This factor appears as (r - 1)¹ in the second denominator and does not appear in the other denominators. The highest power is 1.
  3. (r - 5): This factor appears as (r - 5)¹ in the third denominator and is not present in the other denominators. The highest power is 1.
  4. (r + 1): This factor appears as (r + 1)¹ in the third denominator and does not appear in the other denominators. The highest power is 1.

In this case, each unique factor appears only to the power of 1. However, in other expressions, some factors might appear with higher powers. For instance, if we had a denominator like (r - 4)², then we would need to include (r - 4)² in the LCD. By identifying the highest power of each factor, we ensure that our LCD is comprehensive and accounts for all occurrences of each factor in the denominators.

The highest power of each factor acts as a safeguard, ensuring that the LCD can accommodate all the original denominators. This meticulous approach guarantees that the LCD we construct will be correct and effective for subsequent operations with the rational expressions.

Step 4: Construct the LCD

With all the unique factors and their highest powers identified, the final step in finding the least common denominator is to construct the LCD itself. This involves multiplying together each unique factor raised to its highest power. Let's recap our findings:

  • Unique Factors: (r - 4), (r - 1), (r - 5), (r + 1)
  • Highest Powers: Each factor has a highest power of 1.

Now, we multiply these factors together to form the LCD:

LCD = (r - 4)¹ * (r - 1)¹ * (r - 5)¹ * (r + 1)¹

Simplifying this expression, we get:

LCD = (r - 4)(r - 1)(r - 5)(r + 1)

This expression, (r - 1)(r - 4)(r - 5)(r + 1), is the least common denominator for the given rational expressions. It is the smallest expression that is divisible by each of the original denominators. When we look at the options provided, we see that option A, (r-1)(r-4)(r-5), is missing the factor (r + 1). Therefore, option A is not the correct LCD.

Let's examine the other options:

  • Option B: (r - 1)(r + 4)(r - 5) - This is incorrect because it includes (r + 4) instead of (r - 4) and is missing the factor (r + 1).
  • Option C: (r - 1)(r + 1)(r - 4)(r - 5) - This is the correct LCD as it includes all the necessary factors.
  • Option D: (r - 1)(r + 1)(r + 4)(r - 5) - This is incorrect because it includes (r + 4) instead of (r - 4).

Thus, the correct LCD is (r - 1)(r + 1)(r - 4)(r - 5). This constructed LCD serves as the common denominator needed for operations such as adding or subtracting the given rational expressions. The process of multiplying the unique factors together ensures that the LCD is comprehensive and suitable for further algebraic manipulations.

Conclusion

In summary, finding the least common denominator for rational expressions involves a systematic approach: factoring the denominators, identifying unique factors, determining the highest power of each factor, and constructing the LCD by multiplying these factors together. For the given expressions:

18r−4⋅−5rr2−5r+4⋅r+7r2−4r−5\frac{18}{r-4} \cdot \frac{-5r}{r^2-5r+4} \cdot \frac{r+7}{r^2-4r-5}

the correct LCD is (r - 1)(r + 1)(r - 4)(r - 5).

By following these steps, you can confidently find the LCD for any set of rational expressions, which is a crucial skill for simplifying and solving algebraic problems involving fractions. The least common denominator acts as the foundation for combining rational expressions, allowing for seamless addition, subtraction, and further algebraic manipulations. Mastering this skill opens the door to more complex problem-solving in algebra and beyond.