Subtracting Rational Expressions: A Step-by-Step Guide

Hey guys! Let's dive into the world of subtracting rational expressions. It might sound a bit intimidating at first, but trust me, with a little practice, you'll be acing these problems in no time. We're going to break down the process step by step, making it super easy to follow. Our goal is to make sure you not only understand how to subtract these expressions but also why the process works the way it does. We will walk through the specific example of subtracting the rational expressions $\frac{x^2+8 x}{3 x^2+3 x}-\frac{x^2+2 x-6}{3 x^2+3 x}$ . So, grab your pencils, and let's get started!

Understanding the Basics: What are Rational Expressions?

Okay, before we jump into subtraction, let's make sure we're all on the same page about what rational expressions even are. Think of them as the algebraic cousins of fractions. A rational expression is simply a fraction where the numerator and denominator are both polynomials. Remember, a polynomial is an expression made up of variables, coefficients, and non-negative integer exponents, connected by addition, subtraction, and multiplication.

For example, $\frac{x+2}{x-1}$ and $\frac{x^2 - 4}{2x}$ are both rational expressions. The key thing to remember is that you'll have variables in both the top (numerator) and the bottom (denominator) of the fraction. And just like regular fractions, you can't divide by zero! So, when working with rational expressions, we always need to be aware of any values of the variable that would make the denominator equal to zero. These values are called the excluded values, and we'll talk about those a bit later. To be successful in the process of subtracting rational expressions, you should be familiar with the following concepts: factoring polynomials, simplifying fractions, and finding a common denominator. If you are not familiar with the concepts, don't worry, we'll review the key steps required to successfully subtract rational expressions.

Now, let's consider our example $\frac{x^2+8 x}{3 x^2+3 x}-\frac{x^2+2 x-6}{3 x^2+3 x}$. Both expressions are fractions where the numerator and denominator are polynomials. Thus, they are rational expressions.

Step 1: Check for a Common Denominator

Alright, here's the first crucial step: check if the rational expressions already have a common denominator. This is the key to making the subtraction process straightforward. Luckily for us, in our example, $\frac{x^2+8 x}{3 x^2+3 x}-\frac{x^2+2 x-6}{3 x^2+3 x}$, the denominators are identical: $3x^2 + 3x$.

If the denominators aren't the same, then you'll need to find the least common denominator (LCD). We'll touch on how to do that later on in the process. But for now, since we already have a common denominator, we can move right along to the next step. Having a common denominator simplifies everything because we can simply combine the numerators over that shared denominator, just like you would with regular fractions. Remember, the denominator is the same in both expressions, so we can keep the denominator the same, and all we have to do is focus on combining the numerators. This is the beauty of having a common denominator, it streamlines the entire process, making the subtraction much more manageable.

In our particular example, since the denominators are identical, the next steps will be much easier to execute. We will avoid having to find the LCD, or multiplying the numerators and denominators by clever forms of one to get a common denominator. This significantly reduces the chances of making a mistake. So, let's keep things moving, and celebrate that we have a common denominator.

Step 2: Subtract the Numerators

Now that we've confirmed we have a common denominator, it's time for the main event: subtracting the numerators. We'll keep the common denominator and perform the subtraction operation on the expressions in the numerator. In our example, we have:

$\frac{x^2+8 x}{3 x^2+3 x}-\frac{x^2+2 x-6}{3 x^2+3 x}$

Since we have a common denominator, we can simply subtract the numerators and keep the same denominator. This gives us:

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

Notice the parentheses! It's super important to include those, especially when subtracting a polynomial with multiple terms. The parentheses remind us to distribute that negative sign across all the terms in the second numerator. Failing to do this is a common mistake that can easily lead to the wrong answer. So, be careful! If you do miss the parentheses, you may miss a sign change in the problem, and end up with the wrong result. The subtraction of the numerators is all that is required in this step. Once we have subtracted the numerators, we will move on to the next step, which deals with simplifying the fraction.

Step 3: Simplify the Resulting Expression

After subtracting the numerators, we often end up with a new rational expression that can be simplified. This is where we look for opportunities to factor the numerator and denominator and cancel out any common factors. Factoring is the key to simplifying rational expressions. Let's work through our example. First, we distribute the negative sign in the numerator:

$\frac{x^2+8x-x^2-2x+6}{3x^2+3x}$

Then, we combine like terms in the numerator:

$\frac{6x+6}{3x^2+3x}$

Now, we factor both the numerator and the denominator. In the numerator, we can factor out a 6, and in the denominator, we can factor out $3x$:

$\frac{6(x+1)}{3x(x+1)}$

Notice that we now have a common factor of $(x + 1)$ in both the numerator and the denominator. We can cancel these out:

$\frac{6(x+1)}{3x(x+1)} = \frac{6}{3x}$

Finally, we can simplify the fraction $\frac{6}{3x}$ by dividing both the numerator and denominator by 3:

$\frac{6}{3x} = \frac{2}{x}$

So, the simplified form of our original expression is $\frac{2}{x}$. But before we say we're completely done, we need to address something important.

Step 4: Identify Excluded Values

Remember those excluded values we talked about earlier? This is where they come in. Excluded values are the values of the variable that would make the original denominator equal to zero. We need to identify these because a fraction with a denominator of zero is undefined.

To find the excluded values, go back to the original denominator (before any simplification) and set it equal to zero. In our case, the original denominator was $3x^2 + 3x$. So, we solve the equation:

$3x^2 + 3x = 0$

Factor out $3x$: $3x(x + 1) = 0$

This equation is true if either $3x = 0$ or $x + 1 = 0$. Solving for $x$, we get $x = 0$ or $x = -1$. Therefore, the excluded values for our expression are $x = 0$ and $x = -1$. We exclude these values because they would make the original denominator zero, leading to an undefined expression. The excluded values are very important to keep track of, as they define the domain of the expression.

It is important to find the excluded values from the original denominator, and not the simplified version. Although the simplified version may have a different denominator, the excluded values come from the original, where we have not performed any cancellation. We must consider the original function.

Step 5: State the Final Answer

And now, the grand finale! We state our final answer, including the simplified expression and the excluded values. For our example, the final answer is:

$\frac{2}{x}$, where $x \ne 0, -1$

This means that the simplified form of the expression is $\frac{2}{x}$, but we must remember that $x$ cannot be equal to 0 or -1. These values are excluded from the domain of the simplified expression because they would have made the original expression undefined. Always make sure to state these excluded values; otherwise, your answer isn't complete!

Dealing with Different Denominators (Finding the LCD)

What happens if the rational expressions don't have a common denominator? That's when you'll need to find the least common denominator (LCD). Here's a quick rundown of how to do it:

1. Factor each denominator completely.
2. Identify all the unique factors.
3. For each factor, take the highest power that appears in either denominator.
4. Multiply these factors together to get the LCD.

Once you have the LCD, you'll need to rewrite each rational expression with the LCD as its denominator. This usually involves multiplying the numerator and denominator of each fraction by the same expression (a clever form of 1) to get the desired denominator. This may also require some extra steps, which is why having a common denominator makes everything easier. Let's walk through an example.

Suppose we want to subtract: $\frac{1}{x+2} - \frac{2}{x}$.

Our first step is to identify the LCD. Since the denominators $x$ and $(x+2)$ have no common factors, our LCD will be the product of the two terms. The LCD is $x(x+2)$.

Now, we multiply each fraction by a clever form of 1 to give it the LCD. We multiply the first expression by $\frac{x}{x}$, and we multiply the second expression by $\frac{x+2}{x+2}$.

$\frac{1}{x+2} * \frac{x}{x} - \frac{2}{x} * \frac{x+2}{x+2}$

This simplifies to:

$\frac{x}{x(x+2)} - \frac{2(x+2)}{x(x+2)}$

Then, we subtract the numerators and keep the common denominator:

$\frac{x-2(x+2)}{x(x+2)}$

Distributing the negative sign in the numerator, we have:

$\frac{x-2x-4}{x(x+2)}$

Combining like terms, we have:

$\frac{-x-4}{x(x+2)}$

Conclusion: Practice Makes Perfect!

Subtracting rational expressions might seem a bit challenging at first, but with practice, you'll become a pro. Remember the key steps: check for a common denominator, subtract the numerators, simplify the result, and identify excluded values. Don't be afraid to work through lots of examples and ask for help when you need it. Keep practicing, and you'll be subtracting rational expressions like a boss in no time! So, go out there and conquer those fractions! Good luck, and happy subtracting!