Solving Rational Expressions Quotient Step By Step

Hey everyone! Today, we're diving deep into the world of rational expressions, and we're going to tackle a common question that often pops up: figuring out the quotient when you're dividing these expressions. So, let's jump right in and break down the problem step by step, making sure everyone's on the same page.

The Question at Hand

Our mission, should we choose to accept it (and we do!), is to find the quotient of the following rational expressions:

$$\frac{2x + 5}{3x} \div \frac{2x - 1}{2x + 1}$$

Before we even think about diving into the answer choices, let's equip ourselves with the knowledge and tools we need to solve this problem like pros. Remember, it's not just about getting the right answer; it's about understanding the how and why behind it. So, stick with me, and let's make some math magic happen!

Diving Deep into Rational Expressions

Before we even think about dividing, let's break down what we're working with. Rational expressions, at their heart, are simply fractions where the numerator and the denominator are polynomials. Think of them as the algebraic cousins of regular numerical fractions, but instead of just numbers, we've got variables and expressions thrown into the mix.

The beauty (and sometimes the beast) of rational expressions lies in their ability to represent a wide range of mathematical relationships. They pop up in all sorts of contexts, from calculus to physics, and mastering them is a key step in your mathematical journey. So, buckle up, because we're about to make these expressions our new best friends.

The Golden Rule of Dividing Rational Expressions

Alright, guys, here's the golden rule you absolutely need to remember: Dividing by a fraction is the same as multiplying by its reciprocal. Say that five times fast! This is the secret sauce that makes dividing rational expressions not just manageable, but dare I say, even fun. Okay, maybe not fun for everyone, but definitely less intimidating.

So, what does this mean in practice? Well, when we're faced with a division problem like the one we have, we're going to flip the second fraction (the one we're dividing by) and then change the division sign to a multiplication sign. It's like a mathematical magic trick, turning a division problem into a multiplication problem right before our eyes!

Applying the Rule to Our Problem

Let's take our original problem:

$$\frac{2x + 5}{3x} \div \frac{2x - 1}{2x + 1}$$

And apply our golden rule. We flip the second fraction, the (2x - 1)/(2x + 1), to its reciprocal, which is (2x + 1)/(2x - 1). Then, we change the division sign to a multiplication sign. Our problem now looks like this:

$$\frac{2x + 5}{3x} \times \frac{2x + 1}{2x - 1}$$

See? We've already made progress, and we haven't even broken a sweat yet. Now, we're dealing with multiplication, which is often a bit easier to handle than division. We're on the right track!

Multiplying Rational Expressions The Next Step

Now that we've transformed our division problem into a multiplication problem, let's talk about how to actually multiply rational expressions. The process is pretty straightforward, and it mirrors how you'd multiply regular fractions: multiply the numerators together and multiply the denominators together. Simple as that!

Putting It into Practice

Looking at our transformed problem:

$$\frac{2x + 5}{3x} \times \frac{2x + 1}{2x - 1}$$

We're going to multiply the numerators (2x + 5) and (2x + 1), and then we're going to multiply the denominators 3x and (2x - 1). This gives us:

$$\frac{(2x + 5)(2x + 1)}{3x(2x - 1)}$$

We're one step closer to the solution! Now, we just need to handle those multiplications in the numerator and the denominator. Get ready to put your polynomial multiplication skills to the test!

Expanding the Numerator and Denominator

To expand the numerator, we'll use the FOIL method (First, Outer, Inner, Last) to multiply the two binomials (2x + 5) and (2x + 1). This is a classic technique for multiplying binomials, and it ensures we don't miss any terms.

• First: 2x * 2x = 4x^2
• Outer: 2x * 1 = 2x
• Inner: 5 * 2x = 10x
• Last: 5 * 1 = 5

Adding these terms together, we get:

4x^2 + 2x + 10x + 5

Combining like terms (2x and 10x), we simplify the numerator to:

4x^2 + 12x + 5

Now, let's tackle the denominator. We need to distribute the 3x across the terms in (2x - 1):

3x * 2x = 6x^2

3x * -1 = -3x

So, the denominator becomes:

6x^2 - 3x

Putting it all together, our expression now looks like this:

$$\frac{4x^2 + 12x + 5}{6x^2 - 3x}$$

We've expanded both the numerator and the denominator. Give yourself a pat on the back – you're doing great!

Simplifying the Expression The Final Touch

We've done the heavy lifting of multiplying the rational expressions, but before we declare victory, we need to make sure our answer is in its simplest form. This often means looking for opportunities to factor and cancel out common factors in the numerator and denominator.

Factoring The Key to Simplification

Factoring is like the reverse of expanding. Instead of multiplying terms together, we're trying to break down an expression into its constituent factors. This can help us identify common factors that we can cancel out, simplifying our expression.

Let's start by looking at our numerator:

4x^2 + 12x + 5

This is a quadratic expression, and factoring it might take a little bit of effort. We're looking for two binomials that, when multiplied together, give us this quadratic. After some trial and error (or using your favorite factoring technique), we find that:

4x^2 + 12x + 5 = (2x + 5)(2x + 1)

Now, let's turn our attention to the denominator:

6x^2 - 3x

Here, we can factor out a common factor of 3x:

6x^2 - 3x = 3x(2x - 1)

So, our expression now looks like this:

$$\frac{(2x + 5)(2x + 1)}{3x(2x - 1)}$$

Spotting Opportunities for Cancellation

Now comes the moment we've been waiting for! We need to see if there are any common factors in the numerator and the denominator that we can cancel out. A careful look reveals that, unfortunately, there are no common factors we can cancel in this case. The numerator has factors of (2x + 5) and (2x + 1), while the denominator has factors of 3x and (2x - 1). None of these match up.

This means our expression is already in its simplest form. We can't simplify it any further.

The Answer Revealed

After all our hard work, we've arrived at the simplified quotient of our rational expressions:

$$\frac{4x^2 + 12x + 5}{6x^2 - 3x}$$

Or, in its factored form:

$$\frac{(2x + 5)(2x + 1)}{3x(2x - 1)}$$

Now, let's look back at the answer choices provided in the original question:

A. $\frac{-2x - 5}{3x}$ B. $\frac{4x + 4}{5x + 1}$ C. $\frac{4x^2 + 8x - 5}{6x^2 + 3x}$ D. $\frac{4 x^2+12 x+5}{6 x^2-3 x}$

Comparing our result with the answer choices, we can see that option D, $\frac{4 x^2+12 x+5}{6 x^2-3 x}$, matches our simplified expression perfectly!

Wrapping Up

Wow, we've covered a lot of ground! We started with a division problem involving rational expressions, and we've walked through the steps of transforming it into a multiplication problem, expanding the numerator and denominator, factoring, and simplifying. We've even identified the correct answer choice. You guys have nailed it!

Remember, the key to mastering rational expressions is practice. The more you work with them, the more comfortable you'll become with the techniques involved. So, don't be afraid to tackle more problems and challenge yourself. You've got this!

If you want more practice, try changing the signs or coefficients in the original problem and see if you can still arrive at the correct answer. Or, look for similar problems online or in your textbook. The possibilities are endless!

Keep up the great work, and I'll see you in the next math adventure!