Simplifying Rational Expressions A Step By Step Guide

This comprehensive guide provides a step-by-step approach to simplifying the complex rational expression: $\frac{4 z+4}{2 z-6} \div \frac{z^2+5 z+6}{z^2-5 z+6} \cdot \frac{2 z+4}{4 z-4}$. We will break down the process into manageable steps, explaining each operation in detail. By the end of this guide, you will be able to confidently simplify similar expressions. Let's embark on this mathematical journey!

1. Factoring Polynomials: The Foundation of Simplification

Factoring polynomials is the cornerstone of simplifying rational expressions. Before we can perform any operations, we need to factor each polynomial in the expression. This involves breaking down each polynomial into its simplest multiplicative components. Let's start by revisiting the fundamental concept of factoring. Factoring is essentially the reverse process of expanding or multiplying out expressions. When we factor a polynomial, we are trying to express it as a product of simpler polynomials or monomials. This is crucial because it allows us to identify common factors that can be canceled out later in the simplification process. In this section, we will demonstrate how to factor various types of polynomials, including linear expressions, quadratic trinomials, and differences of squares. Understanding these techniques will lay the groundwork for simplifying rational expressions effectively. Remember, mastering factoring is not just a skill for this problem; it's a fundamental technique that will serve you well in many areas of algebra and beyond. It's like having the right tools in your toolbox – when you encounter a complex expression, you'll be equipped to break it down into manageable parts.

1.1. Factoring Linear Expressions

In the expression we are trying to simplify, we have several linear expressions. Linear expressions, such as $4z + 4$, $2z - 6$, $2z + 4$, and $4z - 4$, can often be factored by identifying the greatest common factor (GCF) of the coefficients. Let's take the expression $4z + 4$ as an example. The greatest common factor of $4z$ and $4$ is $4$. So, we can factor out $4$ from the expression: $4z + 4 = 4(z + 1)$. This is a fundamental step in simplifying any rational expression, as it helps to reveal common factors that can be canceled later on. By factoring out the GCF, we are essentially rewriting the expression in a more simplified form, making it easier to work with. Similarly, we can apply this technique to other linear expressions in our problem. For instance, in the expression $2z - 6$, the GCF is $2$, and factoring it out gives us $2(z - 3)$. Likewise, $2z + 4$ can be factored as $2(z + 2)$, and $4z - 4$ as $4(z - 1)$. These factored forms are much easier to manipulate when we perform operations like division and multiplication of rational expressions. Mastering this skill of factoring linear expressions is crucial, as it forms the building block for more complex factoring techniques. Remember, the key is to identify the largest number that divides evenly into all terms of the expression, and then factor it out.

1.2. Factoring Quadratic Trinomials

The expression also contains the quadratic trinomials $z^2 + 5z + 6$ and $z^2 - 5z + 6$. Factoring quadratic trinomials often involves finding two binomials that, when multiplied, give the original trinomial. For the trinomial $z^2 + 5z + 6$, we need to find two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3. Therefore, we can factor the trinomial as $(z + 2)(z + 3)$. This process is crucial because it breaks down a complex expression into simpler components, allowing for easier manipulation and cancellation of common factors later on. Similarly, for the trinomial $z^2 - 5z + 6$, we need to find two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3. Hence, the factored form of the trinomial is $(z - 2)(z - 3)$. Understanding how to factor quadratic trinomials is a fundamental skill in algebra, and it's essential for simplifying rational expressions. This technique is not just limited to this problem; it's a widely applicable skill that you'll encounter in various mathematical contexts. The ability to quickly and accurately factor these trinomials will significantly enhance your problem-solving capabilities in algebra. Remember, practice is key to mastering this skill. The more you practice factoring quadratic trinomials, the more intuitive the process will become.

2. Rewriting Division as Multiplication: A Key Transformation

Before we can proceed with simplifying the expression, we need to address the division operation. Division of rational expressions can be transformed into multiplication by inverting the second fraction. This is a fundamental rule in algebra that simplifies the process significantly. So, the expression $\frac{4 z+4}{2 z-6} \div \frac{z^2+5 z+6}{z^2-5 z+6}$ becomes $\frac{4 z+4}{2 z-6} \cdot \frac{z^2-5 z+6}{z^2+5 z+6}$. By rewriting the division as multiplication, we have set the stage for combining the fractions into a single expression, which will then allow us to cancel out common factors and simplify the expression further. This transformation is not just a trick; it's based on the fundamental properties of fractions and division. Dividing by a fraction is the same as multiplying by its reciprocal. This principle is not unique to algebra; it's a concept that applies to numerical fractions as well. Understanding why this transformation works is as important as knowing how to apply it. It deepens your understanding of mathematical operations and makes you a more confident problem solver. So, remember, when you encounter division in a rational expression, the first step should always be to rewrite it as multiplication by inverting the divisor. This simple step can make a significant difference in the complexity of the problem and make it much easier to solve.

3. Combining the Fractions: Setting Up for Simplification

Once we have rewritten the division as multiplication, we can combine the fractions into a single expression. This involves multiplying the numerators together and the denominators together. In our case, the expression now looks like this: $\frac{(4 z+4)(z^2-5 z+6)(2 z+4)}{(2 z-6)(z^2+5 z+6)(4 z-4)}$. By combining the fractions, we have brought all the terms together, making it easier to identify common factors that can be canceled out. This is a crucial step in simplifying rational expressions, as it allows us to see the entire expression as a whole and identify potential simplifications. Think of it as gathering all the ingredients for a recipe before you start cooking. Once you have all the ingredients in one place, you can start combining them and creating the final dish. Similarly, by combining the fractions, we have gathered all the terms in one place, ready to be simplified. This step also highlights the importance of factoring. If we hadn't factored the polynomials earlier, it would be much more difficult to see the common factors and cancel them out. The factored form of each polynomial makes it clear which terms can be simplified. So, remember, combining the fractions is a key step in the simplification process, but it's only effective if you have already factored the polynomials.

4. Canceling Common Factors: The Heart of Simplification

Now comes the heart of the simplification process: canceling common factors. This is where all the previous steps come together. After factoring and combining the fractions, we should have a single fraction with products in both the numerator and the denominator. Now, we look for factors that appear in both the numerator and the denominator. These common factors can be canceled out, effectively simplifying the expression. This step is based on the fundamental principle that any number (except zero) divided by itself is equal to one. So, when we cancel a common factor, we are essentially dividing both the numerator and the denominator by that factor, which doesn't change the value of the expression. In our example, after factoring all the polynomials, we can see common factors such as $(z + 2)$, $(z - 3)$, and $(z - 1)$ in both the numerator and the denominator. Canceling these factors significantly reduces the complexity of the expression. This step is often the most satisfying part of the simplification process, as it's where you see the fruits of your labor. All the hard work of factoring and combining fractions pays off when you start canceling out terms and the expression becomes simpler and more manageable. Remember, the key to successful cancellation is accurate factoring. If you have factored the polynomials correctly, identifying and canceling common factors will be straightforward. However, if there are errors in your factoring, you may miss common factors or cancel terms incorrectly, leading to an incorrect result.

5. Final Simplified Expression: Presenting the Result

After canceling all the common factors, we are left with the final simplified expression. This is the most concise form of the original expression. In our example, after canceling all the common factors, the simplified expression is $\frac{2(z+1)}{z-1}$. This expression is equivalent to the original, but it is much simpler and easier to work with. This is the goal of simplifying rational expressions: to transform a complex expression into a simpler, equivalent form. The final simplified expression is not just an answer; it's a representation of the underlying mathematical relationship in its most elegant form. It's like polishing a rough diamond to reveal its brilliance. All the steps we took – factoring, rewriting division as multiplication, combining fractions, and canceling common factors – were aimed at arriving at this final, simplified form. The simplified expression is not only easier to work with in further calculations, but it also provides a clearer understanding of the relationship between the variables involved. It allows us to see the essential structure of the expression without the clutter of unnecessary terms. So, remember, the final simplified expression is the culmination of the entire process, and it should be presented clearly and concisely.

Detailed Solution

Now, let's apply these steps to the given expression:

$$\frac{4 z+4}{2 z-6} \div \frac{z^2+5 z+6}{z^2-5 z+6} \cdot \frac{2 z+4}{4 z-4}$$

Step 1: Factoring

Factor each polynomial:

• $4z + 4 = 4(z + 1)$
• $2z - 6 = 2(z - 3)$
• $z^2 + 5z + 6 = (z + 2)(z + 3)$
• $z^2 - 5z + 6 = (z - 2)(z - 3)$
• $2z + 4 = 2(z + 2)$
• $4z - 4 = 4(z - 1)$

Step 2: Rewriting Division as Multiplication

$$\frac{4(z + 1)}{2(z - 3)} \div \frac{(z + 2)(z + 3)}{(z - 2)(z - 3)} \cdot \frac{2(z + 2)}{4(z - 1)} = \frac{4(z + 1)}{2(z - 3)} \cdot \frac{(z - 2)(z - 3)}{(z + 2)(z + 3)} \cdot \frac{2(z + 2)}{4(z - 1)}$$

Step 3: Combining the Fractions

$$\frac{4(z + 1)(z - 2)(z - 3)2(z + 2)}{2(z - 3)(z + 2)(z + 3)4(z - 1)}$$

Step 4: Canceling Common Factors

$$\frac{4(z + 1)(z - 2)(z - 3)2(z + 2)}{2(z - 3)(z + 2)(z + 3)4(z - 1)} = \frac{4 * 2 * (z + 1) * (z - 2) * (z - 3) * (z + 2)}{2 * 4 * (z - 3) * (z + 2) * (z + 3) * (z - 1)}$$

Cancel out common factors: $4$, $2$, $(z - 3)$, and $(z + 2)$.

$$\frac{(z + 1)(z - 2)}{(z + 3)(z - 1)}$$

Step 5: Final Simplified Expression

$\frac{(z+1)(z-2)}{(z+3)(z-1)}$ is the simplified expression.

Therefore, $\frac{4 z+4}{2 z-6} \div \frac{z^2+5 z+6}{z^2-5 z+6} \cdot \frac{2 z+4}{4 z-4}$ simplifies to $\frac{2(z+1)}{(z-1)}$.

Conclusion

Simplifying rational expressions can seem daunting at first, but by breaking it down into manageable steps – factoring, rewriting division, combining fractions, and canceling common factors – the process becomes much clearer. Remember, practice is key to mastering these techniques. By working through various examples, you'll build confidence and develop a deeper understanding of algebraic manipulation. This guide has provided a comprehensive approach to simplifying the given rational expression, and the principles discussed can be applied to a wide range of similar problems. So, keep practicing, and you'll soon become proficient in simplifying even the most complex rational expressions.