Subtracting Rational Expressions: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of rational expressions and tackling a common challenge: subtraction. Specifically, we're going to break down how to solve the following problem:

$\frac{x+8}{x^2-x-12}-\frac{6x+7}{x^2+4x-32}$

This might look a little intimidating at first, but don't worry! We'll take it one step at a time, and by the end of this guide, you'll be a pro at subtracting rational expressions. So, grab your pencils and paper, and let's get started!

1. Factoring the Denominators: The Foundation for Subtraction

The very first thing we need to do when subtracting rational expressions is to factor the denominators. This is crucial because it allows us to identify the least common denominator (LCD), which is essential for combining the fractions. Think of it like subtracting regular fractions – you can't subtract $\frac{1}{2}$ from $\frac{1}{3}$ directly; you need a common denominator like 6.

So, let's factor those denominators:

• Denominator 1: $x^2 - x - 12$

  We're looking for two numbers that multiply to -12 and add up to -1. Those numbers are -4 and 3. Therefore, we can factor this as:

  $x^2 - x - 12 = (x - 4)(x + 3)$

• Denominator 2: $x^2 + 4x - 32$

  Here, we need two numbers that multiply to -32 and add up to 4. These numbers are 8 and -4. So, we have:

  $x^2 + 4x - 32 = (x + 8)(x - 4)$

Now our original expression looks like this:

$\frac{x+8}{(x-4)(x+3)} - \frac{6x+7}{(x+8)(x-4)}$

Why is factoring so important? Factoring breaks down the expressions into their simplest components, making it much easier to identify common factors and, subsequently, the LCD. It's like taking a complicated puzzle and separating the pieces before trying to put it together. By factoring, we reveal the underlying structure of the expressions, which is key to simplifying them.

2. Finding the Least Common Denominator (LCD): The Key to Combining

Now that we've factored the denominators, the next step is to find the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. It's the common ground we need to combine our fractions.

To find the LCD, we look at the factors in each denominator and include each factor the greatest number of times it appears in any one denominator. In our case, the denominators are $(x - 4)(x + 3)$ and $(x + 8)(x - 4)$.

Let's break it down:

• We have the factors $(x - 4)$, $(x + 3)$, and $(x + 8)$.
• The factor $(x - 4)$ appears in both denominators, so we include it once in the LCD.
• The factors $(x + 3)$ and $(x + 8)$ each appear once, so we include them as well.

Therefore, the LCD is:

LCD = $(x - 4)(x + 3)(x + 8)$

Finding the LCD might seem a bit tricky at first, but with practice, it becomes second nature. Remember, the LCD is the key to adding or subtracting fractions – it's the foundation upon which we build our simplified expression. Without a common denominator, we can't combine the fractions, so this step is absolutely essential.

3. Creating Equivalent Fractions: Setting the Stage for Subtraction

With the LCD in hand, our next task is to create equivalent fractions that have the LCD as their denominator. This means we'll be multiplying the numerator and denominator of each fraction by the factors that are missing from its current denominator.

Let's go through each fraction:

• Fraction 1: $\frac{x+8}{(x-4)(x+3)}$

  The denominator is $(x - 4)(x + 3)$. To get the LCD, $(x - 4)(x + 3)(x + 8)$, we need to multiply by $(x + 8)$. So, we multiply both the numerator and denominator by $(x + 8)$:

  $\frac{x+8}{(x-4)(x+3)} \cdot \frac{x+8}{x+8} = \frac{(x+8)(x+8)}{(x-4)(x+3)(x+8)} = \frac{(x+8)^2}{(x-4)(x+3)(x+8)}$

• Fraction 2: $\frac{6x+7}{(x+8)(x-4)}$

  The denominator is $(x + 8)(x - 4)$. To get the LCD, $(x - 4)(x + 3)(x + 8)$, we need to multiply by $(x + 3)$. So, we multiply both the numerator and denominator by $(x + 3)$:

  $\frac{6x+7}{(x+8)(x-4)} \cdot \frac{x+3}{x+3} = \frac{(6x+7)(x+3)}{(x-4)(x+3)(x+8)}$

Now our expression looks like this:

$\frac{(x+8)^2}{(x-4)(x+3)(x+8)} - \frac{(6x+7)(x+3)}{(x-4)(x+3)(x+8)}$

Creating equivalent fractions is a critical step in the subtraction process. By ensuring both fractions have the same denominator, we set the stage for combining the numerators and simplifying the expression. It's like ensuring all the ingredients are properly prepared before you start cooking – it makes the final result much smoother and tastier!

4. Subtracting the Numerators: The Heart of the Operation

With our fractions now sharing a common denominator, we can finally subtract the numerators. This is where the actual subtraction happens, and it's crucial to pay close attention to the signs and distribution.

Our expression is:

$\frac{(x+8)^2}{(x-4)(x+3)(x+8)} - \frac{(6x+7)(x+3)}{(x-4)(x+3)(x+8)}$

We subtract the numerators while keeping the common denominator:

$\frac{(x+8)^2 - (6x+7)(x+3)}{(x-4)(x+3)(x+8)}$

Now, let's expand the numerators:

• $(x + 8)^2 = (x + 8)(x + 8) = x^2 + 16x + 64$
• $(6x + 7)(x + 3) = 6x^2 + 18x + 7x + 21 = 6x^2 + 25x + 21$

Substitute these back into the expression:

$\frac{x^2 + 16x + 64 - (6x^2 + 25x + 21)}{(x-4)(x+3)(x+8)}$

Remember to distribute the negative sign to all terms in the second numerator:

$\frac{x^2 + 16x + 64 - 6x^2 - 25x - 21}{(x-4)(x+3)(x+8)}$

Now, combine like terms in the numerator:

$\frac{(x^2 - 6x^2) + (16x - 25x) + (64 - 21)}{(x-4)(x+3)(x+8)} = \frac{-5x^2 - 9x + 43}{(x-4)(x+3)(x+8)}$

The act of subtracting the numerators is the core of this whole process. It's where we bring the fractions together and see how they interact. By carefully expanding, distributing, and combining like terms, we've simplified the numerator into a single expression, bringing us closer to our final answer.

5. Simplifying the Result: The Final Touches

We've done the heavy lifting, but we're not quite finished yet! The final step is to simplify the result as much as possible. This means checking if the numerator can be factored and if there are any common factors between the numerator and denominator that can be canceled out.

Our expression currently looks like this:

$\frac{-5x^2 - 9x + 43}{(x-4)(x+3)(x+8)}$

First, let's see if we can factor the numerator, $-5x^2 - 9x + 43$. This quadratic doesn't factor easily (or at all, using integers), so we'll leave it as is.

Next, we look for common factors between the numerator and denominator. In this case, there are no common factors. The numerator, $-5x^2 - 9x + 43$, doesn't share any factors with $(x - 4)$, $(x + 3)$, or $(x + 8)$.

Since we can't factor the numerator or cancel any common factors, our simplified expression is:

$\frac{-5x^2 - 9x + 43}{(x-4)(x+3)(x+8)}$

And that's it! We've successfully subtracted the rational expressions and simplified the result. Simplifying is like putting the final touches on a masterpiece – it ensures that our answer is in its most concise and elegant form. While not every expression can be simplified further, it's always worth checking to make sure we've reached the most simplified answer possible.

Conclusion: You've Conquered Subtraction!

Awesome job, guys! You've made it through the process of subtracting rational expressions. Let's recap the key steps:

1. Factor the denominators: Break down the denominators into their simplest factors.
2. Find the LCD: Identify the least common denominator.
3. Create equivalent fractions: Multiply the numerators and denominators to get the LCD.
4. Subtract the numerators: Combine the numerators, paying attention to signs and distribution.
5. Simplify the result: Factor and cancel common factors if possible.

By following these steps, you can confidently tackle any subtraction problem involving rational expressions. Keep practicing, and you'll become a subtraction superstar in no time! Remember, math is like learning a new language – the more you practice, the more fluent you become. So, keep exploring, keep challenging yourself, and most importantly, keep having fun with math!