Simplify The Expression (z^2-4)/(z^2-1) * (z+1)/(z+2) A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem that involves simplifying algebraic expressions. Specifically, we're going to tackle the expression:

$$\frac{z^2-4}{z^2-1} \cdot \frac{z+1}{z+2}$$

Don't worry if it looks intimidating at first glance. We'll break it down step-by-step, so you can follow along easily. By the end of this guide, you'll not only know how to simplify this particular expression but also gain a better understanding of the underlying concepts. So, let's get started!

Understanding the Basics of Simplifying Expressions

Before we jump into the specifics of our problem, let's quickly review the basic principles of simplifying algebraic expressions. At its core, simplification involves rewriting an expression in its simplest form while maintaining its mathematical equivalence. This often involves factoring, canceling common terms, and applying algebraic identities. When you simplify algebraic expressions, you are essentially tidying them up, making them easier to understand and work with. This is a crucial skill in algebra and calculus, as simplified expressions make further calculations much smoother. For instance, imagine trying to solve a complex equation with a messy expression versus solving it with a clean, simplified one – the latter is definitely the way to go!

Factoring plays a significant role in simplifying expressions. It's like taking a number or expression and breaking it down into its building blocks. For example, the number 12 can be factored into 2 x 2 x 3. Similarly, algebraic expressions can be factored into simpler terms. In our problem, we'll be using a special type of factoring known as the difference of squares. Remember the formula: $a^2 - b^2 = (a + b)(a - b)$. This will come in handy when we deal with terms like $z^2 - 4$ and $z^2 - 1$. Factoring allows us to identify common terms in the numerator and denominator of fractions, which we can then cancel out to simplify the expression. Think of it as reducing a fraction to its lowest terms – the same principle applies to algebraic expressions.

Another key concept in simplifying expressions is the idea of canceling common terms. This is where we identify factors that appear in both the numerator and the denominator of a fraction and eliminate them. For example, if we have an expression like $\frac{(x + 2)(x - 1)}{(x + 2)}$, we can cancel out the $(x + 2)$ term from both the top and the bottom, leaving us with $(x - 1)$. This process is valid because we're essentially dividing both the numerator and the denominator by the same quantity, which doesn't change the overall value of the expression. However, it's crucial to remember that we can only cancel out factors, not terms that are added or subtracted within a factor. For instance, we can't cancel out the 'x' in an expression like $\frac{x + 2}{x}$ because 'x' is not a factor of the entire numerator.

Step 1: Factoring the Numerator and Denominator

Alright, let's roll up our sleeves and get into the nitty-gritty of our problem. The first step in simplifying the expression is to factor the numerator and the denominator of the fractions. Factoring is like detective work – we're trying to find the hidden pieces that make up the whole. Remember that difference of squares formula we talked about earlier? That's going to be our trusty tool in this step.

Let's start with the first fraction, $\frac{z^2-4}{z^2-1}$. We need to factor both the numerator ($z^2 - 4$) and the denominator ($z^2 - 1$). Looking at the numerator, we can see that it fits the difference of squares pattern. Here, $z^2$ is like our $a^2$ and 4 is like our $b^2$. So, we can rewrite 4 as $2^2$. Applying the difference of squares formula, $a^2 - b^2 = (a + b)(a - b)$, we get:

$$z^2 - 4 = (z + 2)(z - 2)$$

Great! We've factored the numerator. Now, let's move on to the denominator, $z^2 - 1$. Again, this fits the difference of squares pattern. Here, $z^2$ is our $a^2$ and 1 is our $b^2$ (since $1^2 = 1$). Applying the same formula, we get:

$$z^2 - 1 = (z + 1)(z - 1)$$

Fantastic! We've factored both the numerator and the denominator of the first fraction. Now, let's rewrite the first fraction with its factored form:

$$\frac{z^2-4}{z^2-1} = \frac{(z + 2)(z - 2)}{(z + 1)(z - 1)}$$

Moving on to the second fraction, $\frac{z+1}{z+2}$, we can see that both the numerator ($z + 1$) and the denominator ($z + 2$) are already in their simplest form. There's no further factoring we can do here. Sometimes, things are just nice and simple! Now, let's rewrite the entire expression with the factored form of the first fraction:

$$\frac{z^2-4}{z^2-1} \cdot \frac{z+1}{z+2} = \frac{(z + 2)(z - 2)}{(z + 1)(z - 1)} \cdot \frac{z+1}{z+2}$$

With the expression fully factored, we're one step closer to simplifying it. Factoring is a fundamental skill in algebra, and mastering it will make your life so much easier when dealing with complex expressions and equations. Now that we've factored the expression, the next step is where the magic happens – we'll start canceling out common factors to simplify it further.

Step 2: Canceling Common Factors

Okay, now comes the fun part – canceling out common factors! This is like weeding a garden; we're removing the unnecessary bits to let the main expression shine. In our factored expression,

$$\frac{(z + 2)(z - 2)}{(z + 1)(z - 1)} \cdot \frac{z+1}{z+2}$$

we can see some terms that appear both in the numerator and the denominator. Remember, we can only cancel out factors that are multiplied, not terms that are added or subtracted within a factor. Looking closely, we have $(z + 2)$ in the numerator of the first fraction and in the denominator of the second fraction. These guys can cancel each other out! It's like dividing $(z + 2)$ by $(z + 2)$, which equals 1, so they essentially disappear from the expression.

Similarly, we have $(z + 1)$ in the denominator of the first fraction and in the numerator of the second fraction. These also cancel each other out for the same reason. So, after canceling these common factors, our expression looks much cleaner:

$$\frac{\cancel{(z + 2)}(z - 2)}{\cancel{(z + 1)}(z - 1)} \cdot \frac{\cancel{z+1}}{\cancel{z+2}} = \frac{z - 2}{z - 1}$$

Isn't that satisfying? We've managed to eliminate quite a bit of clutter. Canceling common factors is a crucial step in simplifying expressions because it reduces the expression to its most basic form. By identifying and removing these common factors, we make the expression easier to work with and understand. This is especially important when dealing with more complex problems, where simplifying along the way can save you a lot of time and effort.

It's worth noting that when we cancel out factors, we're essentially dividing both the numerator and the denominator by the same quantity. As long as that quantity is not zero, this operation is valid. However, it's important to keep in mind the values of $z$ that would make the canceled factors equal to zero. In our case, we canceled out $(z + 2)$ and $(z + 1)$. So, $z$ cannot be equal to -2 or -1 because that would make the original expression undefined (division by zero is a big no-no in mathematics!). These are called restrictions on the variable, and it's a good practice to identify them when simplifying expressions.

Step 3: Stating the Simplified Expression

Alright, we've reached the final step! After all the factoring and canceling, we're left with a much simpler expression. Let's recap what we've done so far. We started with:

$$\frac{z^2-4}{z^2-1} \cdot \frac{z+1}{z+2}$$

We factored the numerator and denominator of the first fraction, which gave us:

$$\frac{(z + 2)(z - 2)}{(z + 1)(z - 1)} \cdot \frac{z+1}{z+2}$$

Then, we canceled out the common factors $(z + 2)$ and $(z + 1)$, which led us to:

$$\frac{z - 2}{z - 1}$$

This is our simplified expression! It's much cleaner and easier to work with than the original one. So, the final answer is:

$$\frac{z - 2}{z - 1}$$

But wait, there's one more thing we need to consider. Remember those restrictions on the variable we talked about? We need to state them along with our simplified expression. We identified that $z$ cannot be equal to -2 or -1 because those values would make the original expression undefined. So, the complete answer is:

$$\frac{z - 2}{z - 1}, \quad z \neq -2, \quad z \neq -1$$

Stating the restrictions on the variable is crucial because it ensures that our simplified expression is equivalent to the original expression for all valid values of $z$. It's like adding a disclaimer to our answer, making sure we're not stepping into any mathematical pitfalls.

In this final step, we've not only arrived at the simplified expression but also emphasized the importance of considering restrictions on the variable. Simplifying expressions is not just about getting to the simplest form; it's also about being mindful of the conditions under which the simplification is valid. By stating the restrictions, we ensure the accuracy and completeness of our answer. So, always remember to check for restrictions when simplifying expressions, especially those involving fractions.

Conclusion

And there you have it! We've successfully simplified the expression $\frac{z^2-4}{z^2-1} \cdot \frac{z+1}{z+2}$ to $\frac{z - 2}{z - 1}$, with the restrictions $z \neq -2$ and $z \neq -1$. We walked through the process step-by-step, from factoring the numerator and denominator to canceling common factors and stating the simplified expression along with its restrictions. This is a typical problem that you might encounter in algebra, and mastering these simplification techniques is essential for tackling more advanced math topics. You guys did great!

Simplifying algebraic expressions is a fundamental skill in mathematics, and it's something that you'll use time and time again. Whether you're solving equations, working with functions, or delving into calculus, the ability to simplify expressions will make your mathematical journey much smoother. The key to becoming proficient in simplification is practice. The more you practice, the more comfortable you'll become with the different techniques and the more easily you'll be able to recognize patterns and apply the appropriate steps. So, don't be afraid to tackle challenging problems and make mistakes along the way – that's how we learn and grow!

Remember, math isn't just about memorizing formulas and procedures; it's about understanding the underlying concepts and developing problem-solving skills. By breaking down complex problems into smaller, manageable steps, you can tackle even the most daunting mathematical challenges. So, keep practicing, keep exploring, and keep enjoying the beauty and power of mathematics! And as always, if you have any questions or need further clarification, don't hesitate to ask. Happy simplifying!