Simplifying Rational Expressions A Step By Step Guide
Introduction
In the realm of algebra, rational expressions play a crucial role, often appearing in various mathematical contexts. A rational expression is essentially a fraction where the numerator and denominator are polynomials. Simplifying these expressions is a fundamental skill that allows us to work with them more efficiently and understand their behavior better. This article will delve into the process of simplifying a specific rational expression, providing a detailed, step-by-step guide to help you master this technique. We will specifically address the expression (r^2 - 4r + 5) / (r - 4) - (r^2 + 2r - 8) / (r - 4). By understanding the underlying principles and techniques, you'll be well-equipped to tackle a wide range of rational expression simplification problems. This comprehensive guide aims to enhance your algebraic skills and boost your confidence in handling complex expressions.
Understanding Rational Expressions
Before we dive into the specific problem, let's clarify what a rational expression is and why simplification is important. At its core, a rational expression is a fraction where both the numerator and the denominator are polynomials. Polynomials, you'll recall, are expressions involving variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples include x^2 + 3x - 2, 5y - 1, and even simple constants like 7. When we form a fraction using two such polynomials, we get a rational expression. Now, the importance of simplification cannot be overstated. A simplified rational expression is easier to work with, making it less prone to errors when performing further operations like addition, subtraction, multiplication, or division. Simplified expressions also offer a clearer picture of the expression's behavior, including its domain (the set of permissible input values) and any potential discontinuities (points where the expression is undefined). For instance, consider the rational expression (2x + 4) / (x + 2). It might appear complex at first glance, but with a simple factorization in the numerator, we can rewrite it as 2(x + 2) / (x + 2). This simplification immediately reveals that the (x + 2) term cancels out, leaving us with just 2, provided x ≠-2 (since division by zero is undefined). This example underscores the power of simplification in making expressions more manageable and revealing their true nature.
The Problem: (r^2 - 4r + 5) / (r - 4) - (r^2 + 2r - 8) / (r - 4)
Now, let's turn our attention to the specific problem we aim to solve: (r^2 - 4r + 5) / (r - 4) - (r^2 + 2r - 8) / (r - 4). This expression involves the subtraction of two rational expressions. A key observation here is that both expressions share the same denominator, (r - 4). This common denominator is a significant advantage, as it allows us to combine the numerators directly. Think of it like subtracting fractions with the same denominator – we simply subtract the numerators and keep the denominator. However, before we rush into the subtraction, it's crucial to remember the order of operations and the importance of distributing the negative sign correctly. Failing to do so can lead to errors in the final result. Our goal is to simplify this expression as much as possible, which means combining like terms, factoring if possible, and ultimately arriving at the most concise form of the rational expression. The process may involve several algebraic manipulations, but by following a systematic approach, we can confidently navigate through the steps and arrive at the correct simplified form. In the following sections, we will break down each step in detail, providing explanations and justifications for each operation.
Step 1: Combining the Numerators
The first crucial step in simplifying the given rational expression is to combine the numerators. Since both fractions share a common denominator, (r - 4), we can directly subtract the numerators. This process is analogous to subtracting regular fractions with a common denominator, where we simply subtract the numerators and keep the denominator the same. However, it's paramount to pay close attention to the negative sign preceding the second fraction. This negative sign applies to the entire numerator of the second fraction, meaning we need to distribute it carefully to each term within the numerator. This is a common area for errors, so meticulousness is key. When we combine the numerators, we get: (r^2 - 4r + 5) - (r^2 + 2r - 8). Now, the next step involves distributing the negative sign to the terms inside the second parenthesis. This means changing the signs of each term: r^2 becomes -r^2, +2r becomes -2r, and -8 becomes +8. Once we've correctly distributed the negative sign, we can rewrite the expression as: r^2 - 4r + 5 - r^2 - 2r + 8. This expression is now ready for the next step, which involves combining like terms.
Step 2: Simplifying the Numerator by Combining Like Terms
After combining the numerators and distributing the negative sign, our expression looks like this: r^2 - 4r + 5 - r^2 - 2r + 8. The next step is to simplify this expression by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have r^2 terms, r terms, and constant terms. Let's identify and group them together: We have r^2 and -r^2, which are like terms. We also have -4r and -2r, which are like terms, and finally, we have the constants 5 and 8, which are also like terms. Now, we can combine these like terms by adding their coefficients. The r^2 terms, r^2 and -r^2, cancel each other out (1r^2 - 1r^2 = 0). For the r terms, we have -4r - 2r, which gives us -6r. Lastly, for the constant terms, we have 5 + 8, which equals 13. Therefore, after combining like terms, the numerator simplifies to -6r + 13. This simplified numerator is a significant step forward, as it reduces the complexity of the expression. We now have a much cleaner expression to work with, making it easier to identify any further simplifications or potential factorizations. In the next step, we will put this simplified numerator back into the rational expression and see if we can simplify the entire expression further.
Step 3: Expressing the Simplified Rational Expression
Having simplified the numerator to -6r + 13, we can now express the entire rational expression in its simplified form. Remember, the original expression was (r^2 - 4r + 5) / (r - 4) - (r^2 + 2r - 8) / (r - 4). We combined the numerators, distributed the negative sign, and then simplified by combining like terms. This process led us to the simplified numerator of -6r + 13. Now, we simply place this simplified numerator back over the common denominator, which is (r - 4). This gives us the simplified rational expression: (-6r + 13) / (r - 4). At this point, it's crucial to examine the expression to see if any further simplification is possible. This often involves checking if the numerator and denominator share any common factors that can be canceled out. Factoring is a key technique here, as it allows us to identify these common factors. However, in this particular case, the numerator (-6r + 13) is a linear expression and doesn't lend itself to easy factorization. Similarly, the denominator (r - 4) is also a simple linear expression. A quick inspection reveals that there are no common factors between the numerator and the denominator. Therefore, we can confidently conclude that the expression (-6r + 13) / (r - 4) is indeed the fully simplified form of the original rational expression. This final form is much easier to work with and provides a clear representation of the expression's behavior.
Final Answer
After meticulously following the steps of combining numerators, distributing the negative sign, simplifying by combining like terms, and expressing the result in its simplest form, we have arrived at the final answer. The simplified rational expression for (r^2 - 4r + 5) / (r - 4) - (r^2 + 2r - 8) / (r - 4) is (-6r + 13) / (r - 4). This result matches option A, which is the correct answer. Throughout this process, we've emphasized the importance of careful algebraic manipulation, paying close attention to details like distributing negative signs and combining like terms accurately. These skills are fundamental in algebra and are crucial for success in more advanced mathematical topics. Understanding how to simplify rational expressions not only helps in solving specific problems but also provides a deeper understanding of algebraic structures and their properties. This knowledge will serve as a solid foundation for tackling more complex mathematical challenges in the future.
