Simplifying Rational Expressions How To Solve (x+1)(x-1)/(x+7)(x+1)

Hey guys! Today, we're diving into the world of rational expressions and how to simplify them. It might sound intimidating, but trust me, it's like solving a puzzle! We'll break down the steps and tackle a specific example to make it crystal clear. So, buckle up and let's get started!

Understanding Rational Expressions

Before we jump into simplifying, let's quickly define what a rational expression actually is. Think of it as a fraction, but instead of numbers, we've got polynomials in the numerator and denominator. Polynomials are expressions with variables and coefficients, like x^2 + 3x - 5 or 2x + 1. So, a rational expression looks something like this: (x^2 + 3x - 5) / (2x + 1). The key here is that the denominator cannot be zero, because division by zero is a big no-no in math. That's why we often see restrictions like x ≠ -1 or x ≠ -7, as in our example question. These restrictions tell us which values of x would make the denominator zero, and we need to avoid those.

Now, why do we even bother simplifying these expressions? Well, just like simplifying regular fractions (like reducing 4/8 to 1/2), simplifying rational expressions makes them easier to work with. It can help us solve equations, graph functions, and perform other mathematical operations more efficiently. Think of it as cleaning up a messy room – once everything is organized, it's much easier to find what you need!

The process of simplifying rational expressions boils down to finding common factors in the numerator and denominator and then canceling them out. This is similar to simplifying numerical fractions where you find common factors between the numerator and the denominator and divide them out. For example, with the fraction 15/25, you recognize that both 15 and 25 have a common factor of 5. Dividing both by 5 gives you the simplified fraction 3/5. The same principle applies to rational expressions, but instead of numerical factors, we're looking for polynomial factors. Factoring polynomials is a crucial skill here. We need to be comfortable with techniques like factoring out a greatest common factor (GCF), factoring quadratic expressions, and recognizing special patterns like the difference of squares. Once we've factored the numerator and denominator, we can identify any factors that appear in both, and these are the ones we can cancel out. It's like finding matching puzzle pieces – once you see them, you can fit them together (or in this case, cancel them out) to simplify the whole picture. And remember, those restrictions we talked about earlier? They're super important! We need to keep track of them throughout the simplification process, because even though a factor might disappear during simplification, it still represents a value that would make the original expression undefined.

Question Breakdown: rac{(x+1)(x-1)}{(x+7)(x+1)}

Let's look at the question we're tackling today:

$\frac{(x+1)(x-1)}{(x+7)(x+1)}$

And the question asks which of the following options is equal to this rational expression when $x \neq -1$ or -7?

A. $\frac{x+1}{x+7}$

B. $\frac{x-1}{x+7}$

C. $\frac{x+7}{x-1}$

First, notice the conditions: $x \neq -1$ or -7. This is crucial! It means x cannot be -1 or -7 because that would make the denominator zero, which is a big no-no in math. These are the values that make the original expression undefined. Think of it like this: we're working with a delicate equation, and these values are like stepping on a crack – they'll break it! So, we need to keep these restrictions in mind as we simplify.

Now, let's break down the expression itself. We have a fraction with polynomials in both the numerator and the denominator. The numerator is (x+1)(x-1), and the denominator is (x+7)(x+1). The first thing we want to do is look for common factors. This is like spotting matching socks in your laundry pile – you know they belong together! In this case, we see that both the numerator and the denominator have a factor of (x+1). This is our matching sock! This is the key to simplifying this rational expression.

Identifying Common Factors: The expression already helpfully has the numerator and the denominator in factored form. This makes our job much easier! We can clearly see the factors: (x+1), (x-1), and (x+7). Spotting the common factor of (x+1) is like finding the secret ingredient in a recipe – it's what will make the simplification work. If the expression wasn't already factored, our first step would be to factor both the numerator and the denominator. This might involve techniques like factoring out a greatest common factor, factoring quadratic expressions, or using special factoring patterns like the difference of squares. Factoring is a fundamental skill in algebra, and it's essential for simplifying rational expressions. Think of it as breaking down a complex problem into smaller, more manageable pieces. Once we have the expression factored, identifying common factors becomes much easier.

Simplifying the Expression: Step-by-Step

This is where the magic happens! We've identified the common factor, now we can cancel it out. It's like dividing both the numerator and the denominator by the same number in a regular fraction. Canceling the (x+1) term from both the numerator and the denominator leaves us with:

$\frac{(x-1)}{(x+7)}$

Canceling Common Factors: Canceling common factors is the heart of simplifying rational expressions. It's based on the fundamental principle that dividing any non-zero expression by itself equals 1. So, when we cancel (x+1) from both the numerator and denominator, we're essentially multiplying the expression by (x+1)/(x+1), which is equal to 1 (as long as x isn't -1). This doesn't change the value of the expression, but it does make it simpler. Think of it as taking out a common thread from a woven fabric – you're removing something that's present in both the top and bottom layers, leaving a cleaner, simpler design. It's crucial to remember that we can only cancel factors, not terms. Factors are things that are multiplied together, while terms are things that are added or subtracted. For example, in the expression (x+1)/(x^2+1), we cannot cancel the 1s because they are terms, not factors. This is a common mistake, so it's important to be careful! The ability to correctly identify and cancel common factors is what allows us to transform a complex rational expression into its simplest form.

The Simplified Form: So, after canceling the common factor, we're left with $\frac{x-1}{x+7}$. This is the simplified form of the original expression. It's like taking a tangled mess of wires and neatly organizing them – the underlying connection is still the same, but it's much easier to see and understand. Notice that this matches option B from our choices. But hold on! We're not quite done yet. We need to remember those restrictions we talked about earlier.

Remembering the Restrictions

Even though we canceled out the (x+1) term, the restriction $x \neq -1$ still applies. It's like a ghost from the original expression that we can't forget! The original expression was undefined when x = -1, and even though our simplified expression looks perfectly fine when x = -1, we have to remember the original restriction. This is a crucial step in simplifying rational expressions. We're not just changing the way the expression looks, we're also preserving its mathematical meaning. The simplified expression is equivalent to the original expression for all values of x except those that make the original expression undefined. It's like saying,