Simplifying (2x+14)/(x^2+6x-7) * (7x-7)/(6x+12) A Step-by-Step Guide

Rational expressions can often appear daunting at first glance, especially when they involve complex fractions and polynomials. However, with a systematic approach, simplifying these expressions can become a manageable task. This comprehensive guide will walk you through the process of simplifying the expression (2x+14)/(x^2+6x-7) * (7x-7)/(6x+12), providing clear explanations and step-by-step instructions along the way. Whether you are a student learning about rational expressions for the first time or someone looking to refresh your skills, this article will equip you with the knowledge and confidence to tackle similar problems. Our goal is to break down each step, ensuring that you understand the underlying principles and can apply them to other algebraic challenges. We will cover factoring, identifying common factors, and reducing the expression to its simplest form. By the end of this guide, you will not only know how to simplify this specific expression but also have a solid foundation for working with rational expressions in general. So, let’s dive in and transform this seemingly complex problem into a series of straightforward steps. Remember, practice is key, and with each problem you solve, your understanding and skills will grow. This guide is designed to be thorough and accessible, making the process of simplifying rational expressions clear and intuitive.

1. Factoring Polynomials: The Foundation of Simplification

Before we can simplify the given rational expression, factoring polynomials is a crucial first step. Factoring breaks down complex expressions into simpler, manageable parts. This process is akin to finding the prime factors of a number, but in the realm of algebra. We'll start by factoring each polynomial in the expression (2x+14)/(x^2+6x-7) * (7x-7)/(6x+12). Factoring not only simplifies the expression but also reveals common factors that can be canceled out later, making the simplification process much easier. Think of factoring as the foundation upon which the rest of our simplification process is built. Without it, we would be trying to simplify a complex jumble of terms, but with it, we can methodically reduce the expression to its simplest form. Let's begin by examining the numerator 2x+14. We look for the greatest common factor (GCF) that can be factored out. In this case, both terms are divisible by 2, allowing us to rewrite the numerator as 2(x+7). Next, we'll tackle the denominator x^2+6x-7. This is a quadratic expression, and we look for two numbers that multiply to -7 and add to 6. These numbers are 7 and -1, so we can factor the denominator as (x+7)(x-1). This step is pivotal, as it sets the stage for identifying common factors between the numerator and the denominator. Factoring transforms complex polynomials into products of simpler terms, enabling us to see the structure more clearly. Now, let's move on to the second fraction's numerator, 7x-7. The GCF here is 7, so we can factor it as 7(x-1). Lastly, we factor the second fraction's denominator, 6x+12. The GCF here is 6, and we factor it as 6(x+2). With each polynomial factored, we are one step closer to simplifying the entire expression. Remember, the goal of factoring is to break down each part into its simplest components, making it easier to identify and cancel common factors. Now that we have factored each polynomial, we can rewrite the entire expression in its factored form, setting the stage for the next step: canceling common factors.

2. Rewriting the Expression with Factored Forms

Once we've factored each polynomial, the next critical step is to rewrite the entire expression using these factored forms. This makes it easier to visualize and identify common factors that can be canceled out. By rewriting the expression, we transform it from a seemingly complex fraction into a more manageable form, where the individual components are clearly visible. This step is like assembling the pieces of a puzzle; each factored polynomial is a piece, and rewriting the expression puts them all together in a way that reveals the overall picture. Let's start by substituting the factored forms we found in the previous section into the original expression (2x+14)/(x^2+6x-7) * (7x-7)/(6x+12). We factored 2x+14 as 2(x+7), x^2+6x-7 as (x+7)(x-1), 7x-7 as 7(x-1), and 6x+12 as 6(x+2). Replacing the original polynomials with their factored forms, we get the expression [2(x+7)]/[(x+7)(x-1)] * [7(x-1)]/[6(x+2)]. This rewritten form is a pivotal transition; it allows us to clearly see the structure of the expression and identify the factors that appear in both the numerators and the denominators. Without this step, it would be much harder to spot the common factors and simplify the expression effectively. Rewriting the expression is not just a matter of substitution; it's about creating a visual representation that makes the simplification process more intuitive. The factored form highlights the relationships between the terms and makes it easier to apply the principles of fraction simplification. Think of it as translating the expression into a language that is easier to understand and manipulate. Once the expression is rewritten in its factored form, we can move on to the exciting part: canceling out the common factors. This is where the simplification really begins to take shape, and the expression starts to shrink into its simplest form. The next section will guide you through the process of identifying and canceling these common factors, bringing us closer to the final simplified expression. By carefully rewriting the expression, we have set the stage for a smooth and efficient simplification process.

3. Identifying and Canceling Common Factors: The Heart of Simplification

The identification and cancellation of common factors is at the heart of simplifying rational expressions. This process involves looking for factors that appear in both the numerator and the denominator of the expression, and then