LCD Of Rational Expressions: A Step-by-Step Guide

Hey guys! Let's break down how to find the least common denominator (LCD) for the rational expressions $\frac{n}{n^2-1}$ and $\frac{-7n}{n^2+4n-5}$. This is a crucial skill when you're adding or subtracting rational expressions, so let's get right to it!

Understanding the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest multiple that two or more denominators share. When dealing with fractions, especially rational expressions (fractions with polynomials), finding the LCD is essential for performing addition and subtraction. It allows us to combine the fractions easily and accurately. Think of it like finding a common language between different fractional parts so we can work with them together. Without a common denominator, it's like trying to add apples and oranges – it just doesn't work! In essence, the LCD simplifies the process of combining rational expressions, making complex algebraic manipulations much more manageable.

To really understand the importance, consider a simple numerical example. Suppose you want to add $\frac{1}{4}$ and $\frac{1}{6}$. The denominators are 4 and 6. The LCD is 12. We convert both fractions to have this denominator: $\frac{1}{4} = \frac{3}{12}$ and $\frac{1}{6} = \frac{2}{12}$. Now, adding them is straightforward: $\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$. The same principle applies to rational expressions, only with polynomials. By identifying the LCD, we transform the rational expressions into equivalent forms that share a common denominator, enabling us to combine them effectively. Factoring the denominators is a key step in finding the LCD, as it helps us identify the unique factors and their highest powers present in each denominator. This ensures that the LCD includes all necessary factors to accommodate each original denominator. Remember, the LCD must be divisible by each of the original denominators.

Step 1: Factor the Denominators

The first thing we need to do is factor each denominator completely. This will help us identify all the unique factors involved. Factoring is a critical step because it breaks down complex expressions into simpler components, revealing the building blocks of the denominators. By identifying these building blocks, we can determine the LCD more easily. It's like disassembling a machine to see all its individual parts. Let's dive into factoring our denominators:

Factoring $n^2 - 1$

The first denominator is $n^2 - 1$. Recognize this? It's a difference of squares! The difference of squares formula is $a^2 - b^2 = (a - b)(a + b)$. Applying this to our expression, we get:

$n^2 - 1 = (n - 1)(n + 1)$

Factoring $n^2 + 4n - 5$

The second denominator is $n^2 + 4n - 5$. We need to find two numbers that multiply to -5 and add to 4. Those numbers are 5 and -1. So we can factor this quadratic as:

$n^2 + 4n - 5 = (n + 5)(n - 1)$

So, our factored denominators are:

• $n^2 - 1 = (n - 1)(n + 1)$
• $n^2 + 4n - 5 = (n + 5)(n - 1)$

Step 2: Identify Unique Factors

Now that we've factored both denominators, we need to identify all the unique factors present. Unique factors are those that appear in either denominator, without repetition within each denominator. This step is like taking inventory of all the distinct components we need to build our LCD. The goal is to ensure that our LCD includes all necessary factors to be divisible by each original denominator. Remember, if a factor appears multiple times in a single denominator (e.g., $(x+2)^2$), we only consider it once when identifying unique factors, but we take the highest power of that factor when constructing the LCD.

Looking at our factored forms:

• $(n - 1)(n + 1)$
• $(n + 5)(n - 1)$

We can see the unique factors are $(n - 1)$, $(n + 1)$, and $(n + 5)$. Make sure you only list each unique factor once, even if it appears in both denominators. For instance, $(n - 1)$ appears in both, but we only include it once in our list of unique factors.

Step 3: Determine the LCD

The LCD is the product of all the unique factors, each raised to the highest power that appears in any of the denominators. In our case, each unique factor appears only once in each denominator, so we simply multiply them together. Constructing the LCD is like assembling a machine using all the necessary parts. Each unique factor contributes to the LCD, ensuring that it is divisible by each of the original denominators. If a factor appears with different powers in different denominators, we take the highest power to ensure divisibility.

So, the LCD is: $(n - 1)(n + 1)(n + 5)$

This means we need to rewrite each of our original rational expressions with this denominator. Let's keep going!

Step 4: Rewrite the Rational Expressions

Now that we have our LCD, we need to rewrite each rational expression with this new denominator. To do this, we multiply both the numerator and denominator of each fraction by the factors that are missing from its original denominator compared to the LCD. Rewriting the rational expressions is like converting measurements to a common unit so we can perform calculations. By expressing each fraction with the LCD, we ensure that they have a common ground for addition or subtraction. This process involves multiplying both the numerator and denominator by the same factors, which doesn't change the value of the fraction but transforms its appearance.

Rewriting $\frac{n}{n^2 - 1}$

We have $\frac{n}{(n - 1)(n + 1)}$. To get the LCD, $(n - 1)(n + 1)(n + 5)$, we need to multiply the denominator by $(n + 5)$. So, we also multiply the numerator by $(n + 5)$:

$\frac{n}{n^2 - 1} = \frac{n(n + 5)}{(n - 1)(n + 1)(n + 5)} = \frac{n^2 + 5n}{(n - 1)(n + 1)(n + 5)}$

Rewriting $\frac{-7n}{n^2 + 4n - 5}$

We have $\frac{-7n}{(n + 5)(n - 1)}$. To get the LCD, $(n - 1)(n + 1)(n + 5)$, we need to multiply the denominator by $(n + 1)$. So, we also multiply the numerator by $(n + 1)$:

$\frac{-7n}{n^2 + 4n - 5} = \frac{-7n(n + 1)}{(n + 5)(n - 1)(n + 1)} = \frac{-7n^2 - 7n}{(n - 1)(n + 1)(n + 5)}$

Step 5: Combine (If Necessary)

If the problem asked you to add or subtract the rational expressions, you would now combine the numerators over the common denominator. Since we were only asked to find the LCD, we're done after rewriting the expressions. Combining the rational expressions is like merging ingredients in a recipe. Once we have a common denominator, we can simply add or subtract the numerators to obtain the final result. This step is where all the previous work pays off, allowing us to simplify complex expressions into a single, manageable fraction.

If we were to combine these, we would have:

$\frac{n^2 + 5n}{(n - 1)(n + 1)(n + 5)} + \frac{-7n^2 - 7n}{(n - 1)(n + 1)(n + 5)} = \frac{n^2 + 5n - 7n^2 - 7n}{(n - 1)(n + 1)(n + 5)} = \frac{-6n^2 - 2n}{(n - 1)(n + 1)(n + 5)}$

Final Answer

The least common denominator for the rational expressions $\frac{n}{n^2-1}$ and $\frac{-7 n}{n^2+4 n-5}$ is $(n - 1)(n + 1)(n + 5)$.

Hope this helps you guys out! Let me know if you have any other questions. Keep up the great work! You're on your way to mastering rational expressions. Remember, practice makes perfect. The more you work with these concepts, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're part of the learning process. And most importantly, have fun with it! Math can be challenging, but it can also be incredibly rewarding. Keep exploring, keep questioning, and keep learning!