Subtracting Rational Expressions A Step-by-Step Guide

Hey guys! Ever stumbled upon rational expressions and felt like you're decoding a secret language? Well, you're not alone! Rational expressions, those fractions with polynomials, can seem intimidating at first. But trust me, once you grasp the basics, they become much less scary. In this article, we're going to dive deep into a specific problem: finding the difference between the rational expressions 7/(3x²) and (4x+5)/x. We'll break down each step, making sure you understand the logic behind the math. So, grab your pencils, and let's get started on this mathematical adventure!

Understanding Rational Expressions

Before we jump into solving the specific problem, let's take a moment to understand what rational expressions actually are. Think of them as fractions where the numerator and denominator are polynomials. A polynomial is simply an expression involving variables and coefficients, like x², 3x + 2, or even just the number 5. So, a rational expression might look like (x² + 1) / (2x - 3), or even the expressions we're dealing with in this problem: 7/(3x²) and (4x+5)/x.

The key thing to remember when working with rational expressions is that we treat them a lot like regular fractions. We can add them, subtract them, multiply them, and divide them. However, just like with regular fractions, we need to pay attention to things like finding a common denominator when adding or subtracting. And that's exactly what we'll be focusing on in this article. Our main goal here is to really understand how to subtract one rational expression from another, and we'll do this by taking it step-by-step. The core concept revolves around finding that common denominator, which allows us to combine the fractions into a single, simplified expression. We'll also emphasize why each step is necessary, building your understanding from the ground up. So, let's get into the nitty-gritty of how to tackle these expressions. We'll make sure you're not just memorizing steps, but truly understanding the underlying principles at play. This way, you can confidently tackle similar problems in the future.

The Problem: 7/(3x²) - (4x+5)/x

Okay, let's get down to the specific problem we're tackling: finding the difference between 7/(3x²) and (4x+5)/x. In mathematical terms, that's:

7/(3x²) - (4x+5)/x

This might look a bit intimidating at first glance, but don't worry, we're going to break it down into manageable steps. The first key step in subtracting rational expressions is, just like with regular fractions, to find a common denominator. Remember, we can only directly subtract fractions if they have the same denominator. So, our initial focus is going to be on figuring out what that common denominator is for 3x² and x.

Think of it like this: you can't directly subtract apples from oranges, right? You need to find a common unit, like "fruit," to be able to combine them. Similarly, we need a common denominator to combine these rational expressions. The common denominator is essentially the "common unit" for our fractions. It allows us to rewrite the fractions in a way that we can perform the subtraction. Once we have the common denominator, we can then adjust the numerators accordingly and perform the subtraction. This process ensures that we're subtracting equivalent quantities, just expressed in a different form. We'll walk through this process meticulously, making sure you understand how each term is transformed and why. By the end of this section, you'll have a solid grasp of the initial steps involved in subtracting rational expressions and be ready to move on to the next stage of simplifying the result.

Finding the Common Denominator

So, how do we find the common denominator for 3x² and x? Well, the common denominator needs to be something that both 3x² and x can divide into evenly. A simple way to think about it is to look for the least common multiple (LCM) of the denominators. In this case, we're looking for the LCM of 3x² and x.

Let's break this down. The number part is easy: the LCM of 3 and 1 (since x is the same as 1x) is simply 3. Now, let's look at the variable part. We have x² and x. The LCM here needs to include the highest power of x that appears in either denominator. So, we need x². Putting it all together, the least common denominator (LCD) for 3x² and x is 3x². This means that 3x² is the smallest expression that both 3x² and x can divide into without leaving a remainder. Why is this important? Because it allows us to rewrite our original fractions with the same denominator, making subtraction possible.

Now, let's think about why the LCD is so crucial. Imagine trying to subtract fractions with wildly different denominators – it would be a mess! The LCD provides a standardized base for our fractions, allowing us to perform the subtraction in a meaningful way. It's like converting different units of measurement to the same unit before adding or subtracting them. For instance, you wouldn't try to add meters and centimeters without first converting them to the same unit. Similarly, the LCD allows us to work with rational expressions in a consistent manner. This step of finding the LCD is often the trickiest part for many students, but once you've mastered it, the rest of the process becomes much smoother. So, make sure you're comfortable with this concept before moving on – it's the foundation for success in this type of problem. We'll use this LCD in the next section to rewrite our original expressions, setting the stage for the actual subtraction.

Rewriting the Expressions with the Common Denominator

Now that we've found our least common denominator (LCD) to be 3x², we need to rewrite both of our rational expressions with this denominator. Remember our original problem:

7/(3x²) - (4x+5)/x

The first expression, 7/(3x²), already has the denominator we want, so we can leave it as is. That's convenient, right? No extra work needed there! But the second expression, (4x+5)/x, needs some adjusting. We need to figure out what to multiply the denominator, x, by to get our LCD, 3x².

Think about it: what do we multiply x by to get 3x²? We need to multiply by 3x. But here's the golden rule of fractions: whatever you do to the denominator, you must do to the numerator! This ensures that we're not changing the value of the fraction, just its appearance. So, we'll multiply both the numerator (4x+5) and the denominator x by 3x. This gives us (3x * (4x+5)) / (3x * x), which simplifies to (12x² + 15x) / (3x²). Now, both of our expressions have the same denominator, 3x², which means we're ready to perform the subtraction!

Let's really zoom in on why we multiply both the numerator and the denominator by the same thing. It's all about maintaining the fraction's value. Multiplying by 3x/3x is essentially multiplying by 1, since any number (except 0) divided by itself is 1. Multiplying by 1 doesn't change the value, it just changes the way the fraction looks. This is a crucial concept in fraction manipulation, and it's the reason why we can confidently rewrite fractions with different denominators without altering their fundamental value. It's like changing a five-dollar bill into five one-dollar bills – you still have the same amount of money, just in a different form. So, by rewriting (4x+5)/x as (12x² + 15x) / (3x²), we've created an equivalent fraction that allows us to perform the subtraction with ease. Next, we'll delve into the actual subtraction process, where we'll combine the numerators over the common denominator.

Performing the Subtraction

Alright, we've done the groundwork. We've found the common denominator, 3x², and we've rewritten our expressions. Now comes the fun part: actually subtracting the fractions! Remember, we're trying to solve:

7/(3x²) - (4x+5)/x

And we've rewritten it as:

7/(3x²) - (12x² + 15x)/(3x²)

Now that the denominators are the same, we can combine the numerators. We simply subtract the second numerator from the first, keeping the common denominator. This gives us:

(7 - (12x² + 15x)) / (3x²)

Notice the parentheses around (12x² + 15x). This is super important! We're subtracting the entire expression, not just the first term. This means we need to distribute the negative sign to both terms inside the parentheses. So, let's rewrite this, carefully distributing the negative:

(7 - 12x² - 15x) / (3x²)

Now we've got a single fraction with a numerator that we can rearrange to look a little neater. Let's put the terms in descending order of their exponents:

(-12x² - 15x + 7) / (3x²)

And there you have it! We've successfully subtracted the rational expressions. This is the core of the problem, the actual execution of the subtraction. Let's think for a moment about why distributing the negative sign is so critical. It's a common mistake to forget this step, but it can completely change the answer. Subtracting an entire expression is like taking away a whole package – you need to take away everything inside the package, not just the first item. The parentheses are there to remind us that (12x² + 15x) is a single unit being subtracted. By distributing the negative, we ensure that we're subtracting each term correctly. The rest of the process – combining the numerators and rearranging the terms – is relatively straightforward once you've mastered this key concept. So, always remember those parentheses when subtracting expressions! In the next section, we'll take a final look at our result and compare it to the given answer choices.

Comparing to the Answer Choices

Okay, we've arrived at our solution:

(-12x² - 15x + 7) / (3x²)

Now, let's take a look at the answer choices provided and see if we can find a match.

A. (-12x² + 15x + 7) / (3x²) B. (-12x² - 15x + 7) / (3x²) C. (-4x + 12) / (3x²) D. (-4x + 2) / (3x²)

Comparing our solution to the choices, we can see that option B, (-12x² - 15x + 7) / (3x²), matches perfectly. Hooray! We've successfully solved the problem.

But let's not just stop there. It's always a good idea to take a moment and reflect on the process. Why is it important to compare our final answer to the given choices? Well, it's a crucial step in verifying our work. It helps us catch any potential errors we might have made along the way. If our answer doesn't match any of the choices, it's a red flag that we need to go back and review our steps. Maybe we made a mistake in distributing the negative sign, or perhaps we miscalculated the common denominator. Comparing our answer forces us to think critically about our solution and ensures that we're confident in our result. It's like double-checking your work before submitting an important assignment – it's a safeguard against careless mistakes. And in mathematics, even small errors can lead to incorrect answers. So, make it a habit to always compare your solution to the available choices, if any, and use it as an opportunity to validate your understanding of the problem.

Key Takeaways and Practice Tips

Awesome! We've successfully navigated the world of rational expressions and solved our problem. Let's recap the key steps we took:

1. Understanding Rational Expressions: We defined what rational expressions are and how they relate to regular fractions.
2. Finding the Common Denominator: We identified the least common denominator (LCD) as the crucial first step.
3. Rewriting the Expressions: We multiplied the numerator and denominator of one fraction to achieve the LCD.
4. Performing the Subtraction: We carefully subtracted the numerators, remembering to distribute the negative sign.
5. Comparing to Answer Choices: We verified our solution against the given options.

Now, what are some tips for mastering these types of problems? First and foremost, practice, practice, practice! The more you work with rational expressions, the more comfortable you'll become with the process. Start with simpler problems and gradually increase the complexity. Pay close attention to the details, especially when distributing negative signs and combining like terms. And don't be afraid to break down problems into smaller, more manageable steps. Why is practice so important? Because it's not enough to just understand the concepts – you need to be able to apply them fluently and accurately. Practice helps you develop that fluency and identify any areas where you might be struggling. It also reinforces the underlying principles and helps you build confidence in your problem-solving abilities. Think of it like learning a musical instrument – you can read all about the techniques, but you won't become proficient until you actually start playing. Similarly, with rational expressions, the key to mastery lies in consistent practice. So, grab some practice problems, put your newfound knowledge to the test, and watch your skills soar!

Conclusion

So, there you have it! We've successfully tackled the challenge of subtracting rational expressions. Remember, the key is to break down the problem into smaller steps, focus on finding the common denominator, and pay close attention to those pesky negative signs. With practice and patience, you'll be a pro at rational expressions in no time! Keep up the great work, and don't hesitate to ask for help when you need it. You've got this!

Question

What is the difference of the rational expressions below?

rac{7}{3 x^2}-rac{4 x+5}{x}

A. rac{-12 x^2+15 x+7}{3 x^2} B. rac{-12 x^2-15 x+7}{3 x^2} C. rac{-4 x+12}{3 x^2} D. rac{-4 x+2}{3 x^2}

Solution

The correct answer is B. $\frac{-12 x^2-15 x+7}{3 x^2}$

To find the difference between the given rational expressions, we need to subtract $\frac{4x + 5}{x}$ from $\frac{7}{3x^2}$. Here’s a step-by-step solution:

1. Find the Least Common Denominator (LCD)

The denominators are $3x^2$ and $x$. The LCD is the least common multiple of these two denominators, which is $3x^2$.

2. Rewrite the Expressions with the LCD

• The first expression, $\frac{7}{3x^2}$, already has the LCD as its denominator, so we don't need to change it.

• For the second expression, $\frac{4x + 5}{x}$, we need to multiply both the numerator and the denominator by $3x$ to get the LCD:

  $\frac{4x + 5}{x} \times \frac{3x}{3x} = \frac{3x(4x + 5)}{3x^2} = \frac{12x^2 + 15x}{3x^2}$

3. Subtract the Expressions

Now we subtract the second expression from the first:

$\frac{7}{3x^2} - \frac{12x^2 + 15x}{3x^2}$

Since they have the same denominator, we can combine the numerators:

$\frac{7 - (12x^2 + 15x)}{3x^2}$

4. Simplify the Numerator

Distribute the negative sign:

$\frac{7 - 12x^2 - 15x}{3x^2}$

Rearrange the terms in the numerator to match the standard form of a polynomial:

$\frac{-12x^2 - 15x + 7}{3x^2}$

Thus, the difference between the rational expressions is:

$\frac{-12x^2 - 15x + 7}{3x^2}$