Simplify Rational Expressions: A Step-by-Step Guide

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Hey guys! Today, we're going to dive into the exciting world of rational expressions and learn how to simplify them like pros. Specifically, we'll tackle a division problem that involves factoring, canceling, and ultimately arriving at the simplest form. So, buckle up and let's get started!

Understanding the Problem

Before we jump into the solution, let's take a closer look at the problem we're trying to solve. We have the following division:

x2+4x−52x2÷x−14x\frac{x^2+4 x-5}{2 x^2} \div \frac{x-1}{4 x}

Our goal is to perform this division and simplify the resulting expression as much as possible. This involves a few key steps, including factoring, inverting and multiplying, canceling common factors, and stating any restrictions on the variable x. Understanding these steps is crucial for mastering the art of simplifying rational expressions. This expression looks a bit intimidating at first glance, right? But don't worry, we'll break it down step by step. The first thing we need to remember is that dividing by a fraction is the same as multiplying by its reciprocal. This simple trick is the key to unlocking the solution. Remember that rational expressions are essentially fractions where the numerator and denominator are polynomials. Just like with regular fractions, we can perform operations like addition, subtraction, multiplication, and division. However, there are a few extra things we need to keep in mind, such as factoring and identifying restrictions on the variable. These restrictions are values of x that would make the denominator equal to zero, which is a big no-no in mathematics. So, keep an eye out for those as we move forward!

Step 1: Factoring the Polynomials

The first step in simplifying this expression is to factor the polynomials in the numerators and denominators. This will help us identify common factors that we can cancel out later. Let's start with the quadratic expression in the numerator of the first fraction:

x2+4x−5x^2 + 4x - 5

We need to find two numbers that multiply to -5 and add up to 4. Can you think of what they might be? If you guessed 5 and -1, you're absolutely right! So, we can factor the quadratic as follows:

x2+4x−5=(x+5)(x−1)x^2 + 4x - 5 = (x + 5)(x - 1)

Now, let's look at the denominators. The denominator of the first fraction is 2x², which is already in a pretty simple form. The denominator of the second fraction is 4x, which is also straightforward. Factoring is a crucial skill in algebra, and it's especially important when dealing with rational expressions. By factoring polynomials, we can break them down into simpler components, making it easier to identify common factors and simplify the expression. There are various factoring techniques, such as factoring out the greatest common factor (GCF), factoring by grouping, and factoring quadratic expressions. The specific technique you'll use depends on the structure of the polynomial you're trying to factor. Practice makes perfect when it comes to factoring, so don't be afraid to tackle lots of problems! The more you practice, the better you'll become at recognizing patterns and applying the appropriate factoring methods. This step is like prepping our ingredients before we start cooking – we need to break everything down into its simplest parts so we can work with it effectively. Factoring might seem like a small step, but it's a huge key to simplifying these expressions.

Step 2: Invert and Multiply

Remember that dividing by a fraction is the same as multiplying by its reciprocal. So, to perform the division, we'll invert the second fraction and multiply it by the first fraction. This gives us:

x2+4x−52x2÷x−14x=x2+4x−52x2⋅4xx−1\frac{x^2+4 x-5}{2 x^2} \div \frac{x-1}{4 x} = \frac{x^2+4 x-5}{2 x^2} \cdot \frac{4 x}{x-1}

Now that we've factored the polynomials and flipped the second fraction, we're ready to multiply the fractions together. To do this, we simply multiply the numerators and multiply the denominators:

(x+5)(x−1)2x2⋅4xx−1=(x+5)(x−1)(4x)2x2(x−1)\frac{(x + 5)(x - 1)}{2x^2} \cdot \frac{4x}{x - 1} = \frac{(x + 5)(x - 1)(4x)}{2x^2(x - 1)}

Inverting and multiplying is a fundamental rule when dividing fractions, and it applies perfectly to rational expressions as well. This step transforms the division problem into a multiplication problem, which is often easier to handle. Think of it like this: dividing by something is the same as multiplying by its inverse. Just like how dividing by 2 is the same as multiplying by 1/2, dividing by a fraction is the same as multiplying by its flipped version. This might seem like a small trick, but it's a powerful one that simplifies the entire process. It's like using a universal adapter when you're traveling – it allows you to connect to different systems without any hassle. And in the world of rational expressions, inverting and multiplying is that universal adapter that makes division a breeze!

Step 3: Canceling Common Factors

Now comes the fun part: canceling out common factors! We have the expression:

(x+5)(x−1)(4x)2x2(x−1)\frac{(x + 5)(x - 1)(4x)}{2x^2(x - 1)}

Notice that we have an (x - 1) factor in both the numerator and the denominator. We can cancel these out. Also, we can simplify the 4x and 2x² terms. 4 divided by 2 is 2, and x divided by x² is 1/x. This gives us:

(x+5)(x−1)(4x)2x2(x−1)=(x+5)(2)x\frac{(x + 5)(\cancel{x - 1})(4x)}{2x^2(\cancel{x - 1})} = \frac{(x + 5)(2)}{x}

Canceling common factors is like decluttering your workspace – it gets rid of unnecessary elements and makes the expression cleaner and easier to manage. It's important to remember that we can only cancel factors that are multiplied, not terms that are added or subtracted. This is a common mistake, so always double-check before you cancel anything. Think of it like this: you can only cancel out matching pieces in a multiplication puzzle. Each cancellation brings us closer to the simplest form of the expression, like peeling away layers of complexity to reveal the elegant solution underneath. It's a satisfying step that transforms a potentially messy expression into something much more manageable. And who doesn't love a good decluttering session, right?

Step 4: Simplify and State Restrictions

After canceling the common factors, we're left with:

2(x+5)x\frac{2(x + 5)}{x}

We can optionally distribute the 2 in the numerator, but it's often fine to leave it in factored form:

2x+10x\frac{2x + 10}{x}

Now, let's talk about restrictions. Remember that we need to identify any values of x that would make the original denominator equal to zero. Looking back at the original problem, we had denominators of 2x² and x - 1. Setting these equal to zero, we get:

2x² = 0 => x = 0 x - 1 = 0 => x = 1

So, the values x = 0 and x = 1 are not allowed. We need to state these restrictions alongside our simplified expression. Identifying restrictions is like setting boundaries in a relationship – it's crucial for maintaining a healthy and stable mathematical environment. We need to make sure that our expression doesn't try to do something impossible, like dividing by zero. Think of these restrictions as guardrails on a highway, preventing us from driving off the edge. They are just as important as the simplified expression itself, because they tell us where the expression is valid and where it's not. It's like having a map with marked areas to avoid – it ensures we navigate the mathematical landscape safely and effectively. This final step completes our journey, providing not just the simplified answer but also the necessary context for its use.

Final Answer

Therefore, the simplified expression is:

2(x+5)x\frac{2(x + 5)}{x}

with the restrictions x ≠ 0 and x ≠ 1.

And there you have it! We've successfully divided and simplified the given rational expression, identified the restrictions, and arrived at our final answer. Remember, the key to success with these problems is to take them step by step, factoring, inverting and multiplying, canceling common factors, and always stating the restrictions. You got this!

So, the answer to the question is: 2(x+5)

Perform the operation x2+4x−52x2÷x−14x\frac{x^2+4 x-5}{2 x^2} \div \frac{x-1}{4 x} and simplify the result. What is the simplified expression in the form ?(x+□)?

Simplify Rational Expressions: A Step-by-Step Guide