Simplifying Rational Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of rational expressions, and specifically, how to subtract them. We'll be tackling the problem: Subtract: $\frac{x^2-2 x-24}{x^2-3 x-18}-\frac{6}{x+3}$. Don't worry, it might look a bit intimidating at first, but trust me, we'll break it down step by step and make it super easy to understand. We'll work through the process to figure out which of the provided options (A, B, C, or D) is the correct answer. Let's get started!

Understanding the Basics of Rational Expressions

Before we jump into the subtraction, let's quickly recap what rational expressions are. Rational expressions are simply fractions where the numerator and denominator are polynomials. Remember, a polynomial is an expression with variables and coefficients, like $x^2 - 2x - 24$ or $x + 3$. When working with rational expressions, we need to remember a few key rules, much like when dealing with regular fractions. The most important thing is that you can't divide by zero! So, we always need to be aware of any values of 'x' that would make our denominator equal to zero. This is crucial for determining the domain of our expression. We'll touch on this later when we simplify our expression.

Now, let's talk about the specific problem we're going to solve, which involves subtracting two rational expressions. The fundamental principle is much like subtracting regular fractions: you need a common denominator. If the denominators are different, we have to find a way to make them the same. This often involves factoring and multiplying by clever forms of 1. It's like a mathematical puzzle, and once you get the hang of it, it can be quite fun! The key takeaway here is to understand the core concept: We need a common denominator before we can subtract the numerators. Once we have a common denominator, we can subtract the numerators and simplify the resulting expression.

Step-by-Step Solution to the Problem

Alright, let's get down to business and solve the problem. We have the expression $\frac{x^2-2 x-24}{x^2-3 x-18}-\frac{6}{x+3}$. Here's how we'll break it down:

1. Factor the expressions: The first step is to factor both the numerator and the denominator wherever possible. This helps us identify any common factors that might simplify the expression. Let's start with the first fraction, where we have $x^2 - 2x - 24$ in the numerator and $x^2 - 3x - 18$ in the denominator. Let's factor the numerator, we are looking for two numbers that multiply to -24 and add up to -2. Those numbers are -6 and 4. So we can factor the numerator to get $(x-6)(x+4)$. Next, let's factor the denominator. We are looking for two numbers that multiply to -18 and add up to -3. Those numbers are -6 and 3. So we can factor the denominator to get $(x-6)(x+3)$. This gives us $\frac{(x-6)(x+4)}{(x-6)(x+3)} - \frac{6}{x+3}$.

2. Find the common denominator: Now, we need to identify the least common denominator (LCD) for our two fractions. Looking at the factored expressions, the first fraction has $(x-6)(x+3)$ in the denominator and the second has $(x+3)$. The LCD is $(x-6)(x+3)$.

3. Rewrite the fractions with the common denominator: Since the first fraction already has the LCD, we only need to adjust the second fraction. We multiply the second fraction's numerator and denominator by $(x-6)$. This gives us: $\frac{6(x-6)}{(x+3)(x-6)}$. So now we have $\frac{(x-6)(x+4)}{(x-6)(x+3)} - \frac{6(x-6)}{(x+3)(x-6)}$.

4. Combine and simplify the numerators: Now that the fractions share the common denominator, we can subtract the numerators and keep the same denominator. This gives us: $\frac{(x-6)(x+4)-6(x-6)}{(x-6)(x+3)}$. Expanding the numerator gives us $\frac{x^2 - 2x - 24 - 6x + 36}{(x-6)(x+3)}$. Combining like terms, we get $\frac{x^2 - 8x + 12}{(x-6)(x+3)}$.

5. Factor and cancel: We can factor the numerator $x^2-8x+12$ into $(x-6)(x-2)$. Therefore, the expression becomes $\frac{(x-6)(x-2)}{(x-6)(x+3)}$. We can now cancel the common factor $(x-6)$, as long as $x \ne 6$. This simplifies to $\frac{x-2}{x+3}$.

Therefore, the simplified form of the original expression is $\frac{x-2}{x+3}$.

Analyzing the Answer Choices

Now that we've found our answer, let's see which of the provided options matches our simplified expression. We calculated that subtracting the expressions gives us $\frac{x-2}{x+3}$.

• A. $\frac{x^2-2 x-30}{x^2-3 x-18}$: This is not correct because this is not the expression we found after simplifying.
• B. $x-2$: This is not correct because it is not the simplified form of the expression.
• C. $\frac{x+4}{x+3}$: This is not correct because this is not the expression we found after simplifying.
• D. $\frac{x-2}{x+3}$: This matches our calculated result exactly! Therefore, this is the correct answer. The critical step here is to make sure you've factored everything correctly and simplified appropriately.

So, the correct answer is D. $\frac{x-2}{x+3}$. Congratulations, we have solved the problem!

Important Considerations and Domain

Remember how we mentioned the domain of rational expressions earlier? It's essential to consider this when simplifying. The domain refers to all possible values of 'x' that the expression can take. We need to exclude any values that would make the denominator equal to zero, because division by zero is undefined. Let's look back at the original expression, $\frac{x^2-2 x-24}{x^2-3 x-18}-\frac{6}{x+3}$.

Before simplifying, the denominators are $x^2 - 3x - 18$ and $x + 3$. Factoring the first denominator gives us $(x - 6)(x + 3)$. Therefore, the original expression has denominators of $(x - 6)(x + 3)$ and $(x + 3)$. This tells us that $x$ cannot equal 6 or -3, as these values would result in division by zero. After simplifying our expression to $\frac{x-2}{x+3}$, the denominator is $x + 3$. This means $x$ cannot equal -3. Even though the factor of $(x-6)$ canceled out in the simplification, it's critical to remember that in the original expression, 'x' could not equal 6. So, we must exclude the values that make the original denominator equal to zero. The domain of the simplified expression is all real numbers except $x = -3$ and $x = 6$.

Conclusion: Mastering Rational Expression Subtraction

We did it, guys! We successfully subtracted two rational expressions, simplified our answer, and identified the correct choice from the given options. We also learned how to determine the domain of the expression. Remember, practice makes perfect. The more problems you solve, the more comfortable you'll become with these concepts. Keep practicing, and you'll be a rational expression master in no time! Remember to always factor, find the common denominator, combine, and simplify. And don't forget to watch out for those values of 'x' that make your denominator zero! Keep practicing, and you'll be acing those math problems in no time. If you have any questions, feel free to ask! Happy calculating!