Simplifying Rational Expressions Find The Numerator Of The Sum

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Hey guys! Today, we're diving into the fascinating world of rational expressions. We are going to simplify and add a rational expression, which might seem daunting at first, but trust me, it’s like solving a puzzle – super satisfying once you get the hang of it. So, let's break down the problem step by step and make sure everyone’s on board. Our mission? To tackle the expression: (x / (x^2 + 3x + 2)) + (3 / (x + 1)). This looks like a handful, but with a little algebraic magic, we'll simplify it and find the numerator of the simplified sum. In this comprehensive guide, we'll walk you through each stage of simplifying this expression. We'll start by factoring the quadratic expression in the denominator, identifying common denominators, combining the fractions, and finally simplifying the result. By the end of this article, you'll have a solid understanding of how to handle similar problems. Ready to jump in? Let's do this!

The expression we're tackling involves fractions with polynomials, so it falls under the category of rational expressions. These types of problems are common in algebra and calculus, and mastering them is crucial for more advanced math topics. This introduction sets the stage for a detailed walkthrough, ensuring you're well-prepared to follow along and understand each step of the simplification process. We will explore strategies for simplifying rational expressions, including factoring, finding common denominators, and combining like terms. Each step is explained in detail to ensure clarity and comprehension. This guide is designed to be accessible to anyone, regardless of their prior experience with algebra. Whether you're a student looking to improve your grades, a professional needing a refresher, or just someone who enjoys mathematical challenges, you'll find valuable insights and practical techniques here.

Factoring the Denominator: Unlocking the First Step

The first key step in simplifying our expression is to factor the denominator of the first fraction, which is x^2 + 3x + 2. Factoring is like reverse multiplication – we're trying to find two binomials that, when multiplied, give us the original quadratic expression. In this case, we need to find two numbers that add up to 3 (the coefficient of the x term) and multiply to 2 (the constant term). Those numbers are 1 and 2. This is a crucial step because it allows us to identify common factors between the denominators of the two fractions, which is essential for adding them together. By factoring the quadratic expression, we reveal its underlying structure, making it easier to manipulate and simplify the overall expression. Let's think of it like this: imagine you have a complex machine, and you need to fix it. The first thing you'd do is take it apart to see how it works, right? Factoring is kind of like that – it helps us break down the expression into smaller, more manageable pieces so we can understand it better and find the best way to simplify it.

So, we can rewrite x^2 + 3x + 2 as (x + 1)(x + 2). Now, our expression looks like this: (x / ((x + 1)(x + 2))) + (3 / (x + 1)). See how much simpler it's starting to look? Factoring might seem like a small step, but it's a game-changer. It transforms a complex quadratic expression into a product of simpler binomials, making it easier to see how the two fractions relate to each other. This step is not just about finding the factors; it’s about setting the stage for the rest of the solution. By factoring the denominator, we've unlocked the first key to simplifying this expression. We've taken a potentially intimidating problem and made it much more approachable. This is the power of factoring – it turns complexity into clarity. We've now laid the groundwork for finding a common denominator, which is the next crucial step in adding these fractions. Keep up the great work, guys! We're making excellent progress, and each step we take brings us closer to the final simplified expression.

Finding the Common Denominator: The Key to Combining Fractions

Now that we've factored the denominator of the first fraction, we're ready to tackle the next critical step: finding the common denominator. Remember, to add fractions, they need to have the same denominator. Looking at our expression, (x / ((x + 1)(x + 2))) + (3 / (x + 1)), we can see that the denominators are (x + 1)(x + 2) and (x + 1). The least common denominator (LCD) is the smallest expression that both denominators can divide into evenly. In this case, the LCD is (x + 1)(x + 2). To get the second fraction to have this denominator, we need to multiply both its numerator and denominator by (x + 2). It’s like giving the fraction a makeover – we're changing its appearance without actually changing its value. Think of it like this: if you have a recipe that serves four people, and you need to serve eight, you'd double all the ingredients, right? You're changing the quantities, but the recipe itself – the ratio of ingredients – stays the same. Multiplying the numerator and denominator by the same factor is the same idea. We're scaling the fraction, but its value remains unchanged.

So, we multiply the second fraction, (3 / (x + 1)), by ((x + 2) / (x + 2)), which gives us (3(x + 2) / ((x + 1)(x + 2))). Now, both fractions have the same denominator: (x + 1)(x + 2). This is a huge step because it means we can finally add the fractions together. Finding the common denominator is like finding the missing piece of a puzzle – once you have it, everything else falls into place. Without a common denominator, it's like trying to add apples and oranges – they're just not compatible. But now that we have a common denominator, we can treat these fractions like they're speaking the same language. We can combine them, simplify them, and get closer to our final answer. This step might seem a bit technical, but it's absolutely essential for working with rational expressions. It's the foundation upon which we build the rest of the solution. So, great job guys! We've navigated this tricky part, and we're well on our way to simplifying this expression. Next up, we'll combine the fractions and see what happens. Keep the momentum going!

Combining the Fractions: Bringing It All Together

With both fractions now sporting the same denominator, (x + 1)(x + 2), we're at the exciting stage where we can finally combine them! Our expression looks like this: (x / ((x + 1)(x + 2))) + (3(x + 2) / ((x + 1)(x + 2))). To combine fractions with a common denominator, we simply add their numerators and keep the denominator the same. It's like adding slices of the same pizza – if you have one slice and someone gives you three more, you now have four slices. The