Simplifying Rational Expressions: A Step-by-Step Guide

Hey math enthusiasts! Ever feel like rational expressions are a bit of a puzzle? Well, you're not alone! These expressions, which are essentially fractions with polynomials, can seem a bit intimidating at first. But don't worry, we're going to break down how to combine rational expressions like a pro, making it all super clear and easy to understand. We'll be focusing on a specific example, and by the end, you'll have the skills to tackle similar problems with confidence. So, let's dive in and get started! Our goal is to simplify and combine the following rational expressions: $\frac{1}{x^2-5 x+6}+\frac{3}{x^2-x-2}$. This process involves several key steps: factoring, finding the least common denominator (LCD), rewriting the fractions, and finally, combining the numerators. Let's break this down step-by-step. Buckle up, and let's make this math thing fun! We'll start with the first rational expression, $\frac{1}{x^2-5x+6}$, and the second, $\frac{3}{x^2-x-2}$, and then combine them.

Step 1: Factoring the Denominators

Alright, guys, the first thing we always want to do when dealing with rational expressions is factor the denominators. This is super important because it helps us identify any common factors, which we'll need later on. Factoring also helps us find the least common denominator (LCD), which is crucial for combining the fractions. So, let's get our factoring hats on! For the first denominator, $x^2 - 5x + 6$, we need to find two numbers that multiply to 6 and add up to -5. After a little bit of head-scratching, we realize that -2 and -3 fit the bill. So, we can factor $x^2 - 5x + 6$ into $(x - 2)(x - 3)$. Now, let's move on to the second denominator, $x^2 - x - 2$. We need two numbers that multiply to -2 and add up to -1. Here, -2 and 1 do the trick. So, $x^2 - x - 2$ factors into $(x - 2)(x + 1)$. We've successfully factored both denominators! Now we have: $\frac{1}{(x-2)(x-3)} + \frac{3}{(x-2)(x+1)}$. Remember, factoring is like unlocking the hidden structure of the expression. It makes everything easier to work with, like finding the secret passage in an adventure game! Keep going, and you're doing great!

Step 2: Finding the Least Common Denominator (LCD)

Okay, now that we've factored the denominators, it's time to find the least common denominator (LCD). The LCD is the smallest expression that both denominators can divide into evenly. Think of it as the common ground where we can bring both fractions together. To find the LCD, we take all the unique factors from each denominator and multiply them together, using the highest power of each factor if it appears in multiple denominators. In our case, the first denominator is $(x - 2)(x - 3)$, and the second is $(x - 2)(x + 1)$. The unique factors are $(x - 2)$, $(x - 3)$, and $(x + 1)$. Notice that $(x - 2)$ appears in both denominators, but we only need to include it once in our LCD. So, the LCD is $(x - 2)(x - 3)(x + 1)$. This LCD will be the new denominator for both of our fractions when we rewrite them. The LCD is like the ultimate meeting place for our fractions – all the different parts come together to form a united whole! Keep going; you're doing fantastic! This step is a cornerstone in combining the two fractions.

Step 3: Rewriting the Fractions with the LCD

Alright, folks, it's time to rewrite our fractions so they have the same denominator – the LCD we just found, which is $(x - 2)(x - 3)(x + 1)$. This step is all about making sure our fractions are speaking the same language so we can combine them easily. For the first fraction, $\frac{1}{(x-2)(x-3)}$, we need to multiply both the numerator and the denominator by $(x + 1)$ to get the LCD. This gives us $\frac{1(x+1)}{(x-2)(x-3)(x+1)}$, which simplifies to $\frac{x+1}{(x-2)(x-3)(x+1)}$. For the second fraction, $\frac{3}{(x-2)(x+1)}$, we need to multiply both the numerator and the denominator by $(x - 3)$ to get the LCD. This gives us $\frac{3(x-3)}{(x-2)(x-3)(x+1)}$, which simplifies to $\frac{3x-9}{(x-2)(x-3)(x+1)}$. Now we have two new fractions: $\frac{x+1}{(x-2)(x-3)(x+1)} + \frac{3x-9}{(x-2)(x-3)(x+1)}$. See? We've successfully rewritten both fractions with the same denominator. This process is like giving each fraction a makeover so they can match perfectly and be ready to merge. Keep that energy; you're on the right track!

Step 4: Combining the Numerators

Here we are, at the final stretch! Now that our fractions share the same denominator, which is $(x - 2)(x - 3)(x + 1)$, we can combine the numerators. This is the moment we've been working towards – the grand finale! We simply add the numerators together and keep the common denominator. So, we take $(x + 1)$ and add $(3x - 9)$. This gives us $x + 1 + 3x - 9$, which simplifies to $4x - 8$. Now, we put this new numerator over the common denominator: $\frac{4x-8}{(x-2)(x-3)(x+1)}$. We've successfully combined the two fractions into one! It's like combining two ingredients into one delicious dish. It all comes together to make something greater than the sum of its parts. Remember, guys, the key here is the common denominator. It's the foundation upon which we built our combined fraction. You're doing amazing!

Step 5: Simplifying the Result (If Possible)

Almost there! After combining the numerators, the final step is to simplify the result, if possible. This is where we look for any common factors in the numerator and the denominator that we can cancel out. In our combined fraction, we have $\frac{4x-8}{(x-2)(x-3)(x+1)}$. Let's examine the numerator, $4x - 8$. We can factor out a 4, which gives us $4(x - 2)$. Now our fraction looks like this: $\frac{4(x-2)}{(x-2)(x-3)(x+1)}$. See that $(x - 2)$ in both the numerator and the denominator? Bingo! We can cancel them out. After canceling, we're left with $\frac{4}{(x-3)(x+1)}$. And that's it! We've simplified our expression as much as possible. This final step is like the polishing touch, making sure our answer is in its cleanest, most efficient form. Always remember to check for opportunities to simplify. It makes your answer look sharp and ensures you've fully solved the problem. You've done it! You successfully combined and simplified the original rational expressions, making you a math superstar! Pat yourselves on the back, you deserve it!

Conclusion: You Got This!

And there you have it, folks! We've successfully combined and simplified the rational expressions. This process might seem like a lot of steps, but once you practice it, it becomes second nature. Remember to always factor, find the LCD, rewrite the fractions, combine the numerators, and simplify. With a little practice, you'll be able to tackle any rational expression problem that comes your way. Keep practicing, stay curious, and don't be afraid to ask for help when you need it. Math is a journey, and every step you take makes you stronger. You got this, and I'm here to cheer you on every step of the way! Keep up the fantastic work; you are now well-equipped to combine rational expressions like a pro. Keep practicing and keep up the great work! You've learned something new today and that's always something to be proud of. Keep the momentum going and conquer those rational expressions! Keep it up!