Simplifying Rational Expressions: A Step-by-Step Guide

Hey guys! Today, we are going to tackle a fun math problem: finding the difference between two fractions. Specifically, we'll be working with the expression:

$$\frac{4s}{s^2 - 18s + 81} - \frac{36}{s^2 - 18s + 81}$$

Don't worry, it looks trickier than it actually is. We'll break it down step-by-step, so you'll be a pro at this in no time! This guide will help you not only solve this particular problem but also understand the underlying concepts so you can confidently tackle similar problems in the future. We'll start by understanding the problem, then move on to simplifying the expression, and finally present the answer in its simplest form. Remember, math isn't just about getting the right answer; it's about understanding the process and building a strong foundation for more advanced topics. So, let's dive in and make math a little less intimidating and a lot more fun!

Understanding the Problem

Okay, so let's start by making sure we understand what the problem is asking. We need to find the difference between two fractions. The fractions have algebraic expressions in them, which might seem a bit scary, but it just means we'll need to use our algebra skills! Specifically, we're dealing with rational expressions – fractions where the numerator and denominator are polynomials.

Key to solving this problem is recognizing that both fractions share the same denominator: s^2 - 18s + 81. This is a huge clue because it means we can combine the numerators directly. Think of it like subtracting fractions with common denominators, like 3/5 - 1/5. You just subtract the numerators (3 - 1) and keep the same denominator (5), resulting in 2/5. We'll do something similar here, but with algebraic expressions. Before we jump into combining the fractions, let's take a closer look at that denominator. It looks like it might be factorable, and factoring is often a crucial step in simplifying rational expressions. Factoring the denominator can reveal common factors that can be canceled out, leading to a simpler final answer. Plus, a simplified answer is what we're aiming for! So, understanding the structure of the problem – common denominators and the potential for factoring – sets us up for a smooth solution. Remember, always break down complex problems into smaller, manageable steps. That's the key to mastering any math challenge. Now that we've got a good grasp of what we're facing, let's move on to the next step: simplifying the expression.

Simplifying the Expression

Now comes the fun part – actually simplifying the expression! Remember how we noticed that the denominators are the same? That's great news because it means we can go ahead and combine the numerators. So, let's rewrite the problem by combining the numerators over the common denominator:

$$\frac{4s - 36}{s^2 - 18s + 81}$$

Awesome! Now we have a single fraction. But we're not done yet. To express our answer in the simplest form, we need to see if we can factor anything. Factoring is like the magic key to simplifying many algebraic expressions. Let's start by looking at the numerator, 4s - 36. Do you see any common factors? Yep, both terms are divisible by 4! So, we can factor out a 4:

$$4(s - 9)$$

Now, let's turn our attention to the denominator, s^2 - 18s + 81. This looks like a quadratic expression. Specifically, it might be a perfect square trinomial. Do you remember those? A perfect square trinomial factors into the square of a binomial. Let's see if we can rewrite it. We need to find two numbers that add up to -18 and multiply to 81. Think about it... 9 and 9 work, and since we need -18, we'll use -9 and -9. So, we can factor the denominator as:

$$(s - 9)(s - 9) \text{ or } (s - 9)^2$$

Great job! Now we have factored both the numerator and the denominator. Let's rewrite the entire expression with the factored forms:

$$\frac{4(s - 9)}{(s - 9)(s - 9)}$$

Do you see anything we can cancel out? We have a (s - 9) in both the numerator and the denominator! This is where the magic happens. We can cancel out one (s - 9) from the top and one from the bottom, leaving us with:

$$\frac{4}{s - 9}$$

And there you have it! We've simplified the expression as much as we can. We've factored, canceled out common factors, and now we're left with a much cleaner fraction. Remember, the key to simplifying rational expressions is to factor and cancel. Always look for common factors in both the numerator and the denominator. Now, let's move on to presenting our final answer in its simplest form.

Expressing the Answer in Simplest Form

Okay, we've done the hard work of simplifying the expression, and we've arrived at:

$$\frac{4}{s - 9}$$

Now, we need to make sure this is indeed the simplest form and present it clearly as our final answer. In this case, we've already factored both the numerator and the denominator, and we've canceled out all common factors. There are no more simplifications we can make. The numerator, 4, is a constant, and the denominator, s - 9, is a linear expression that cannot be factored further. So, we're good to go! This fraction is in its simplest form.

However, there's one small detail we need to consider: the values of s that would make the denominator zero. Remember, we can't divide by zero in math. So, we need to identify any values of s that would make s - 9 equal to zero. That value is s = 9. This means that our simplified expression is valid for all values of s except for 9. We sometimes call this a restriction on the variable. While it's important to be aware of this restriction, for this particular problem, we're primarily focused on simplifying the expression. So, we can confidently state that our simplified answer is:

$$\frac{4}{s - 9}$$

To present our final answer clearly, we can write it like this:

Final Answer:

$$\frac{4}{s - 9}$$

And that's it! We've successfully found the difference between the two fractions and expressed the answer in its simplest form. We started by understanding the problem, then we simplified the expression by factoring and canceling, and finally, we presented our answer clearly. Remember, showing your work and presenting the answer clearly are important parts of problem-solving. You not only get the correct solution, but you also communicate your understanding effectively. Now, you're well-equipped to tackle similar problems involving rational expressions. Keep practicing, and you'll become a master of simplification!

Final Answer:

$$\frac{4}{s - 9}$$