Simplify The Rational Expression Z^3+z^2-6 Z-6 Z^2-4 Z-5 A Step By Step Guide
Hey guys! Today, we're diving into the world of rational expressions and tackling a common challenge: simplifying them. Specifically, we're going to break down the expression: $\frac{z3+z2-6 z-6}{z^2-4 z-5}$. Don't worry if it looks intimidating at first glance. We'll take it step by step, and you'll be simplifying like a pro in no time! Before we jump into the nitty-gritty, let's understand what rational expressions are and why simplifying them is so important. A rational expression, at its core, is simply a fraction where the numerator and the denominator are polynomials. Think of it as the algebraic version of a numerical fraction, like 1/2 or 3/4. Just as we simplify numerical fractions to their lowest terms (e.g., reducing 2/4 to 1/2), we simplify rational expressions to make them easier to work with. Simplified expressions are much easier to analyze, graph, and perform operations on, such as addition, subtraction, multiplication, and division. Imagine trying to solve a complex equation with a complicated rational expression – simplifying it first can save you a ton of time and effort! Moreover, simplifying rational expressions helps in identifying any restrictions on the variable. These restrictions are values of the variable that would make the denominator zero, which is undefined in mathematics. Identifying these values is crucial for understanding the domain of the expression and avoiding mathematical errors. So, simplifying rational expressions isn't just about making things look prettier; it's a fundamental skill in algebra and calculus. It's like having a superpower that allows you to see through the complexity and get to the heart of the problem. Now that we've established the importance of simplification, let's get started with our example expression.
Step 1: Factoring the Numerator
Our first task is to factor the numerator, which is the polynomial $z^3 + z^2 - 6z - 6$. Factoring is the process of breaking down a polynomial into simpler expressions (factors) that, when multiplied together, give you the original polynomial. There are several factoring techniques, but for this particular polynomial, we'll use a method called factoring by grouping. Factoring by grouping is a handy technique when you have a polynomial with four terms. The idea is to group the terms in pairs, factor out the greatest common factor (GCF) from each pair, and then see if you can factor out a common binomial factor. Let's apply this to our numerator. First, we group the first two terms and the last two terms: $(z^3 + z^2) + (-6z - 6)$. Now, we factor out the GCF from each group. From the first group, the GCF is $z^2$, and from the second group, the GCF is -6. Factoring these out, we get: $z^2(z + 1) - 6(z + 1)$. Notice that we now have a common binomial factor of $(z + 1)$. This is the key to factoring by grouping! We can factor out $(z + 1)$ from the entire expression: $(z + 1)(z^2 - 6)$. And there you have it! We've successfully factored the numerator. The factored form of $z^3 + z^2 - 6z - 6$ is $(z + 1)(z^2 - 6)$. This factored form is much easier to work with than the original polynomial. Factoring is a crucial skill in simplifying rational expressions, so make sure you're comfortable with different factoring techniques. Practice makes perfect! With the numerator factored, we're one step closer to simplifying our rational expression. Next, we'll tackle the denominator.
Step 2: Factoring the Denominator
Now, let's turn our attention to the denominator of our rational expression, which is the quadratic polynomial $z^2 - 4z - 5$. Factoring quadratic polynomials is a fundamental skill in algebra, and there are several techniques you can use. In this case, we'll use the method of finding two numbers that multiply to the constant term and add up to the coefficient of the linear term. This method works well for quadratics where the leading coefficient (the coefficient of the $z^2$ term) is 1. So, we need to find two numbers that multiply to -5 (the constant term) and add up to -4 (the coefficient of the $z$ term). Let's think about the factors of -5. They are: 1 and -5, or -1 and 5. Which pair adds up to -4? It's 1 and -5! So, we can write the quadratic polynomial as a product of two binomials: $(z + 1)(z - 5)$. And that's it! We've factored the denominator. The factored form of $z^2 - 4z - 5$ is $(z + 1)(z - 5)$. Factoring the denominator is just as crucial as factoring the numerator when simplifying rational expressions. It allows us to identify common factors that can be canceled out, which is the key to simplification. Now that we've factored both the numerator and the denominator, we're ready to put them together and see if we can simplify the entire expression.
Step 3: Rewrite the Expression with Factored Forms
Okay, guys, we've done the hard work of factoring both the numerator and the denominator. Now comes the satisfying part: putting it all together and seeing the simplification magic happen! Remember our original rational expression: $\fracz3+z2-6 z-6}{z^2-4 z-5}$. We factored the numerator as $(z + 1)(z^2 - 6)$ and the denominator as $(z + 1)(z - 5)$. So, we can rewrite the expression using these factored forms{(z + 1)(z - 5)}$. This step is crucial because it visually shows us the common factors that we can cancel out. It's like unveiling the hidden simplicity within the expression. Rewriting the expression in its factored form makes the next step, canceling common factors, much clearer and easier to understand. It's a bridge between the complex original expression and its simplified form. So, take a moment to appreciate the beauty of factoring and how it transforms the expression into a more manageable form. With the expression rewritten, we're now ready to identify and cancel out those common factors. Get ready for the grand finale of simplification!
Step 4: Cancel Common Factors
Alright, the moment we've been waiting for! Now we get to cancel out the common factors in our rational expression. We've rewritten the expression as: $\frac(z + 1)(z^2 - 6)}{(z + 1)(z - 5)}$. Looking at this, do you see any factors that appear in both the numerator and the denominator? That's right! We have a common factor of $(z + 1)$. Canceling out common factors is like dividing both the numerator and the denominator by the same value. Just as we can simplify the numerical fraction 6/8 by dividing both the numerator and denominator by 2, we can simplify rational expressions by canceling common factors. When we cancel out the $(z + 1)$ factor, we're left with{z - 5}$. And there you have it! We've simplified the rational expression. This simplified form is much cleaner and easier to work with than the original expression. However, there's one more important thing we need to consider: restrictions on the variable. Canceling common factors is a powerful simplification technique, but it's crucial to remember that we can only cancel factors that are not equal to zero. In other words, we need to identify any values of z that would make the canceled factor, $(z + 1)$, equal to zero. This leads us to the final step: identifying restrictions.
Step 5: State Restrictions on the Variable
We've successfully simplified our rational expression to $\frac{z^2 - 6}{z - 5}$. But we're not quite done yet! It's super important to state any restrictions on the variable $z$. Restrictions are values of the variable that would make the original denominator equal to zero, which makes the expression undefined. Remember, we canceled out a factor of $(z + 1)$ in the simplification process. This means that $z = -1$ would make the original denominator zero, even though it's not apparent in the simplified form. We also have the factor $(z - 5)$ in the simplified denominator. This means that $z = 5$ would also make the denominator zero. So, our restrictions are $z \neq -1$ and $z \neq 5$. Stating the restrictions is a crucial part of simplifying rational expressions. It ensures that we're working with a valid expression and that we don't inadvertently divide by zero. Think of it as adding a disclaimer to our simplified expression, saying,
