Simplifying Rational Expressions A Step-by-Step Guide

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Hey guys! Today, we're diving into a fun and engaging math problem that involves rational expressions. Don't worry if that sounds intimidating; we're going to break it down step-by-step so it's super easy to follow. Our mission is to simplify the expression: ${\frac{2x}{x-3} - \frac{4x^2}{x^2 + 2x - 15} + \frac{x+1}{x-3}}$. Ready to get started? Let's jump right in!

1. Understanding the Problem

Before we start crunching numbers, let's take a moment to really understand what we're dealing with. Rational expressions are basically fractions where the numerators and denominators are polynomials. In our case, we have three rational expressions that we need to combine. This means we'll need to find a common denominator, perform some algebraic manipulations, and simplify the result. Think of it like putting together a puzzle – each piece (or term) needs to fit just right!

Our expression is:

${ \frac{2x}{x-3} - \frac{4x^2}{x^2 + 2x - 15} + \frac{x+1}{x-3} }$

We've got fractions with polynomials, and our ultimate goal is to make this look as clean and simple as possible. So, stick with me, and we'll conquer this together!

2. Factoring the Denominators

The first crucial step in simplifying rational expressions is to factor the denominators. Why? Because it helps us identify the least common denominator (LCD), which is essential for combining the fractions. Looking at our expression, we notice that the denominator ${x^2 + 2x - 15}$ can be factored. Let's do that now!

We need to find two numbers that multiply to -15 and add up to 2. If you think about it, those numbers are 5 and -3. So, we can factor the quadratic expression as:

${ x^2 + 2x - 15 = (x + 5)(x - 3) }$

Now, our expression looks like this:

${ \frac{2x}{x-3} - \frac{4x^2}{(x + 5)(x - 3)} + \frac{x+1}{x-3} }$

See how much clearer things are becoming? Factoring is like unlocking a secret code that makes the rest of the problem much easier.

3. Identifying the Least Common Denominator (LCD)

Now that we've factored the denominators, we can easily find the least common denominator (LCD). The LCD is the smallest expression that each denominator can divide into evenly. It’s like finding the common ground where all our fractions can meet and play nice.

Looking at our denominators, we have ${x - 3}$ and ${(x + 5)(x - 3)}$. To find the LCD, we need to include each unique factor with the highest power it appears. In this case, we have the factors ${(x - 3)}$ and ${(x + 5)}$. So, the LCD is:

${ LCD = (x + 5)(x - 3) }$

Why is the LCD so important? Because it allows us to rewrite each fraction with the same denominator, making it super easy to add or subtract them. Think of it as converting different currencies into a single currency so you can easily calculate your total expenses.

4. Rewriting Fractions with the LCD

Alright, now for the fun part – rewriting our fractions with the LCD. This step is all about making sure each fraction has the same denominator so we can combine them. It's like making sure everyone is speaking the same language so we can have a coherent conversation.

Our original expression is:

${ \frac{2x}{x-3} - \frac{4x^2}{(x + 5)(x - 3)} + \frac{x+1}{x-3} }$

And our LCD is ${(x + 5)(x - 3)}$. We need to multiply the numerator and denominator of each fraction by the necessary factors to get the LCD.

For the first fraction, ${\frac{2x}{x-3}}$, we need to multiply by ${\frac{x+5}{x+5}}$:

${ \frac{2x}{x-3} \times \frac{x+5}{x+5} = \frac{2x(x+5)}{(x+5)(x-3)} = \frac{2x^2 + 10x}{(x+5)(x-3)} }$

The second fraction, ${\frac{4x^2}{(x + 5)(x - 3)}}$, already has the LCD, so we don't need to change it.

For the third fraction, ${\frac{x+1}{x-3}}$, we also need to multiply by ${\frac{x+5}{x+5}}$:

${ \frac{x+1}{x-3} \times \frac{x+5}{x+5} = \frac{(x+1)(x+5)}{(x+5)(x-3)} = \frac{x^2 + 6x + 5}{(x+5)(x-3)} }$

Now, our expression looks like this:

${ \frac{2x^2 + 10x}{(x+5)(x-3)} - \frac{4x^2}{(x + 5)(x - 3)} + \frac{x^2 + 6x + 5}{(x+5)(x-3)} }$

Great job! We've successfully rewritten all the fractions with the same denominator. Now, we're ready to combine them!

5. Combining the Numerators

Now comes the super satisfying part – combining the numerators! Since all our fractions have the same denominator, we can simply add and subtract the numerators. It's like adding apples to apples – super straightforward!

Our expression looks like this:

${ \frac{2x^2 + 10x}{(x+5)(x-3)} - \frac{4x^2}{(x + 5)(x - 3)} + \frac{x^2 + 6x + 5}{(x+5)(x-3)} }$

Combining the numerators, we get:

${ \frac{(2x^2 + 10x) - 4x^2 + (x^2 + 6x + 5)}{(x+5)(x-3)} }$

Now, let's simplify the numerator by combining like terms:

${ \frac{2x^2 + 10x - 4x^2 + x^2 + 6x + 5}{(x+5)(x-3)} = \frac{(2x^2 - 4x^2 + x^2) + (10x + 6x) + 5}{(x+5)(x-3)} = \frac{-x^2 + 16x + 5}{(x+5)(x-3)} }$

So, our expression now looks like this:

${ \frac{-x^2 + 16x + 5}{(x+5)(x-3)} }$

We're getting closer to the finish line! The next step is to see if we can simplify this fraction any further.

6. Simplifying the Result

Our final step is to simplify the result. This means checking if the numerator can be factored and if there are any common factors between the numerator and the denominator that we can cancel out. It's like putting the final polish on a beautifully crafted piece of work!

Our expression is:

${ \frac{-x^2 + 16x + 5}{(x+5)(x-3)} }$

Let's see if we can factor the numerator, ${-x^2 + 16x + 5}$. Unfortunately, this quadratic expression doesn't factor nicely using integers. We could use the quadratic formula to find the roots, but for the purpose of simplifying the expression, factoring isn't going to help us much here.

Now, let’s check if there are any common factors between the numerator and the denominator. The denominator is ${(x + 5)(x - 3)}$, and the numerator is ${-x^2 + 16x + 5}$. Since we couldn't factor the numerator, there are no common factors to cancel out.

Therefore, our simplified expression is:

${ \frac{-x^2 + 16x + 5}{(x+5)(x-3)} }$

And that's it! We've successfully simplified the rational expression. Give yourself a pat on the back – you've earned it!

7. Conclusion

Wrapping things up, we've successfully simplified the rational expression ${\frac{2x}{x-3} - \frac{4x^2}{x^2 + 2x - 15} + \frac{x+1}{x-3}}$. We tackled this problem by factoring the denominators, finding the least common denominator, rewriting the fractions, combining the numerators, and simplifying the result. Rational expressions might seem tricky at first, but with a systematic approach, they become much more manageable. Remember, the key is to break the problem down into smaller, digestible steps.

Math is all about practice, so don't be afraid to try more problems like this. The more you practice, the more confident you'll become. And hey, if you ever get stuck, just remember our friendly guide here – we've got your back!

Keep up the awesome work, and see you in the next math adventure!