Subtracting Rational Expressions: A Simple Guide

Hey guys! Today, we're going to dive into subtracting rational expressions. It might sound intimidating, but trust me, it's totally manageable. We'll break it down step by step, and by the end, you'll be subtracting fractions with variables like a pro! We're tackling the problem: $\frac{2}{x+2}-\frac{4}{x-2}$, and our goal is to express the answer as a single, simplified, rational expression, leaving the denominator factored. Let's jump right in!

Finding the Common Denominator

The very first thing we always need to do when adding or subtracting fractions – and that includes rational expressions – is to find a common denominator. Think of it like this: you can't directly compare or combine fractions unless they're sliced into the same number of pieces. With regular numerical fractions, we find the least common multiple (LCM) of the denominators. With rational expressions, it's a similar idea, but we're dealing with polynomials instead of numbers. Our denominators here are $(x + 2)$ and $(x - 2)$. These are distinct linear factors, meaning they don't share any common factors (other than 1, of course). This makes our job a little easier! The least common denominator (LCD) is simply the product of these two denominators: $(x + 2)(x - 2)$. This is the expression that both original denominators can divide into evenly. To get each fraction to have this common denominator, we'll need to multiply the numerator and denominator of each fraction by the missing factor from the LCD. For the first fraction, $\frac{2}{x+2}$, we're missing the $(x - 2)$ factor. So, we'll multiply both the top and bottom by $(x - 2)$. For the second fraction, $\frac{4}{x-2}$, we're missing the $(x + 2)$ factor. Therefore, we'll multiply both the numerator and denominator by $(x + 2)$. This process is crucial for ensuring we're working with equivalent fractions, allowing us to combine them correctly. By finding this common denominator, we set the stage for the next step, which involves rewriting the fractions and preparing them for subtraction.

Rewriting the Fractions

Okay, so we've found our common denominator: $(x + 2)(x - 2)$. Now comes the fun part – rewriting our fractions so they both have this denominator. Remember how we talked about multiplying the top and bottom of each fraction by the missing factor? Let's put that into action! For the first fraction, $\frac{2}{x+2}$, we multiply both the numerator and the denominator by $(x - 2)$: $\frac2}{x+2} * \frac{x-2}{x-2} = \frac{2(x-2)}{(x+2)(x-2)}$ Notice that we're not actually performing the multiplication in the denominator yet. We're keeping it in factored form, which will be super helpful later when we simplify. Now, let's do the same for the second fraction, $\frac{4}{x-2}$. This time, we multiply the numerator and denominator by $(x + 2)$ $\frac{4{x-2} * \frac{x+2}{x+2} = \frac{4(x+2)}{(x-2)(x+2)}$ Again, we're leaving the denominator factored. See how both fractions now have the same denominator, $(x + 2)(x - 2)$? Awesome! This means we're ready to combine them. Before we jump into subtracting, though, let's take a moment to appreciate why this step is so important. By rewriting the fractions with a common denominator, we've essentially expressed them in terms of the same “unit.” It's like comparing apples to apples instead of apples to oranges. This allows us to directly subtract the numerators while keeping the denominator consistent. This is a core principle in fraction arithmetic, and it applies equally well to rational expressions.

Performing the Subtraction

Alright, we've got our fractions rewritten with a common denominator. This is where the actual subtraction happens! We now have: $\frac2(x-2)}{(x+2)(x-2)} - \frac{4(x+2)}{(x-2)(x+2)}$ Since the denominators are the same, we can combine the numerators. But, and this is a big but, we need to be super careful with the subtraction sign! It applies to the entire second numerator. Think of it like distributing a negative sign. So, let's rewrite the expression, combining the numerators and being mindful of that subtraction $\frac{2(x-2) - 4(x+2)(x+2)(x-2)}$ Now, we need to simplify the numerator. This involves distributing and combining like terms. First, distribute the 2 in the first term and the -4 (yes, the negative 4!) in the second term $\frac{2x - 4 - 4x - 8(x+2)(x-2)}$ Next, combine the like terms in the numerator $2x$ and $-4x$ combine to $-2x$, and $-4$ and $-8$ combine to $-12$. This gives us: $\frac{-2x - 12{(x+2)(x-2)}$ We're getting closer! The denominator is looking good, all factored and ready to go. But we're not quite done with the numerator yet. There's one more simplification we can make. Let's see what it is.

Simplifying the Expression

Okay, we've arrived at $\frac-2x - 12}{(x+2)(x-2)}$ The denominator is perfectly factored, but let's focus on simplifying the numerator, $-2x - 12$. Notice anything? That's right, there's a common factor! Both terms are divisible by -2. Factoring out the -2 is the key to simplifying this expression further. When we factor out a -2 from the numerator, we get $-2(x + 6)$ So, now our expression looks like this: $\frac{-2(x + 6){(x+2)(x-2)}$ Take a close look at the entire expression. Do we see any common factors between the numerator and the denominator that we can cancel out? Nope! There's no $(x + 6)$ term in the denominator, and we can't cancel anything within the factors $(x + 2)$ or $(x - 2)$. This means we've simplified as much as we can! We've successfully expressed the result as a single, simplified, rational expression. The numerator is factored, and the denominator is factored. We've achieved our goal! This final simplification step is crucial because it ensures that our answer is in its most concise and understandable form. Leaving out this step can sometimes lead to an answer that, while technically correct, isn't fully simplified. By factoring and canceling common factors, we present our solution in the clearest possible way.

Final Answer

So, after all that awesome work, our final answer is: $\frac{-2(x + 6)}{(x+2)(x-2)}$ Give yourselves a pat on the back, guys! We took two rational expressions, found a common denominator, subtracted them, and simplified the result. That's a fantastic accomplishment! Remember, the key to working with rational expressions is to take it one step at a time. Find the common denominator, rewrite the fractions, perform the operation (in this case, subtraction), and then simplify. And always, always double-check your work, especially when dealing with those pesky negative signs! You've now got a solid foundation for tackling more complex rational expression problems. Keep practicing, and you'll become a master of manipulating these expressions. And remember, math is a journey, not a destination. Enjoy the process of learning and problem-solving, and you'll be amazed at what you can achieve. You've got this!