Simplifying Rational Expressions A Step-by-Step Guide

In mathematics, simplifying rational expressions is a fundamental skill, especially in algebra and calculus. This article will thoroughly guide you through the process with an example problem. We aim to provide a detailed, step-by-step solution that clarifies the underlying concepts. Our focus is on enhancing your understanding and problem-solving skills in this area. Whether you're a student, educator, or someone looking to refresh your math knowledge, this guide is designed to help you master the art of simplifying rational expressions.

Understanding Rational Expressions

Before diving into the simplification process, it’s crucial to grasp what rational expressions are. At its core, a rational expression is simply a fraction where the numerator and the denominator are polynomials. Polynomials, as you may recall, are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include 3x^2 + 2x - 1 and 5y - 7. Therefore, a rational expression might look like (3x^2 + 2x - 1) / (5y - 7).

Rational expressions are pervasive in algebra and calculus, appearing in various contexts such as solving equations, graphing functions, and calculus problems involving derivatives and integrals. Mastering the simplification of these expressions is essential for success in higher-level mathematics. When dealing with rational expressions, it's important to remember that the same rules that apply to numerical fractions also apply here. This includes finding common denominators, adding or subtracting numerators, and simplifying the resulting expression.

The process of simplifying rational expressions often involves several algebraic techniques, such as factoring, canceling common factors, and combining like terms. Factoring, in particular, plays a significant role as it allows us to identify common factors in the numerator and the denominator, which can then be canceled out. This step is crucial for reducing the expression to its simplest form. Additionally, understanding the domain of rational expressions is vital. Since division by zero is undefined, we must exclude any values of the variable that make the denominator equal to zero. This consideration is especially important when simplifying expressions, as we need to ensure that the simplified expression is equivalent to the original expression over the same domain.

The Problem

Let's tackle a specific problem to illustrate the process. Our task is to simplify the following expression:

(4w) / (w - 2) + (3w) / (w - 3)

This expression involves the addition of two rational expressions. The denominators, (w - 2) and (w - 3), are different, which means we need to find a common denominator before we can add the fractions. This is a typical scenario in algebra, and the steps we'll follow are applicable to a wide range of similar problems. The key to simplifying this expression lies in understanding how to manipulate fractions with polynomial denominators and applying the rules of algebra correctly.

Step 1: Finding the Common Denominator

The first crucial step in simplifying this expression is to find a common denominator. When adding or subtracting fractions, a common denominator is essential because it allows us to combine the numerators properly. In this case, we have two fractions with denominators (w - 2) and (w - 3). To find the common denominator, we need to identify the least common multiple (LCM) of these two expressions. Since (w - 2) and (w - 3) are distinct linear factors, their least common multiple is simply their product.

Therefore, the common denominator is (w - 2)(w - 3). This means we need to rewrite each fraction with this new denominator. To do this, we multiply the numerator and denominator of each fraction by the factor that is missing from its original denominator. For the first fraction, (4w) / (w - 2), we multiply both the numerator and the denominator by (w - 3). For the second fraction, (3w) / (w - 3), we multiply both the numerator and the denominator by (w - 2). This process ensures that we are not changing the value of the fractions, only their form. Once we have both fractions with the same denominator, we can proceed to add the numerators.

Step 2: Rewriting the Fractions

Now that we've identified the common denominator as (w - 2)(w - 3), we need to rewrite each fraction with this denominator. This involves multiplying the numerator and denominator of each fraction by the appropriate factor. Let's start with the first fraction, (4w) / (w - 2). To get the common denominator, we multiply both the numerator and the denominator by (w - 3):

(4w * (w - 3)) / ((w - 2) * (w - 3))

This gives us:

(4w^2 - 12w) / (w^2 - 5w + 6)

Next, we rewrite the second fraction, (3w) / (w - 3). To get the common denominator, we multiply both the numerator and the denominator by (w - 2):

(3w * (w - 2)) / ((w - 3) * (w - 2))

This simplifies to:

(3w^2 - 6w) / (w^2 - 5w + 6)

Now, both fractions have the same denominator, which allows us to add them together. This step is crucial as it sets the stage for simplifying the expression further.

Step 3: Adding the Fractions

With both fractions now sharing a common denominator, the next step is to add the fractions. This involves adding the numerators while keeping the common denominator. We have:

(4w^2 - 12w) / (w^2 - 5w + 6) + (3w^2 - 6w) / (w^2 - 5w + 6)

To add these fractions, we combine the numerators:

(4w^2 - 12w + 3w^2 - 6w) / (w^2 - 5w + 6)

Now, we simplify the numerator by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have 4w^2 and 3w^2 as well as -12w and -6w. Combining these gives us:

(7w^2 - 18w) / (w^2 - 5w + 6)

This resulting fraction is the sum of the two original fractions. However, we're not done yet. The next step is to check if this fraction can be simplified further.

Step 4: Simplifying the Result

After adding the fractions, we obtained the expression:

(7w^2 - 18w) / (w^2 - 5w + 6)

To simplify this further, we need to see if we can factor both the numerator and the denominator. Factoring involves breaking down a polynomial into simpler expressions that, when multiplied together, give the original polynomial. If we find common factors in the numerator and the denominator, we can cancel them out, which simplifies the expression.

Let's start by factoring the numerator, 7w^2 - 18w. We can factor out a common factor of w:

w * (7w - 18)

Now, let's factor the denominator, w^2 - 5w + 6. We're looking for two numbers that multiply to 6 and add up to -5. Those numbers are -2 and -3. So, we can factor the denominator as:

(w - 2) * (w - 3)

Now our expression looks like this:

(w * (7w - 18)) / ((w - 2) * (w - 3))

In this case, there are no common factors between the numerator and the denominator that can be canceled out. The numerator has factors of w and (7w - 18), while the denominator has factors of (w - 2) and (w - 3). Since there are no matching factors, the expression is already in its simplest form.

Step 5: The Final Answer

After going through the steps of finding a common denominator, rewriting the fractions, adding them, and attempting to simplify the result, we arrive at the simplest form of the expression:

(7w^2 - 18w) / (w^2 - 5w + 6)

Comparing this with the given options, we find that it matches option C.

Therefore, the correct answer is:

C. (7w^2 - 18w) / (w^2 - 5w + 6)

Conclusion

Simplifying rational expressions is a crucial skill in algebra. This step-by-step guide has walked you through the process, from finding common denominators to factoring and canceling terms. Remember, the key is to break down the problem into manageable steps and apply the rules of algebra carefully. By understanding the underlying concepts and practicing regularly, you can master the art of simplifying rational expressions and tackle more complex problems with confidence. Mastering this skill opens doors to more advanced topics in mathematics, making it a worthwhile investment of your time and effort. Whether you're preparing for an exam, working on a challenging assignment, or simply looking to expand your mathematical knowledge, the ability to simplify rational expressions will undoubtedly prove valuable.