Simplifying Rational Expressions A Step-by-Step Guide

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In the realm of mathematics, simplifying rational expressions is a fundamental skill that unlocks doors to more complex algebraic manipulations. Rational expressions, essentially fractions with polynomials in the numerator and denominator, often appear daunting at first glance. However, with a systematic approach and a keen eye for detail, simplifying these expressions becomes an attainable and even enjoyable task. This article will delve into the intricacies of subtracting rational expressions, guiding you through the process with clarity and precision. We will focus on a specific example, but the principles discussed here can be readily applied to a wide range of similar problems. Understanding the core concepts of factoring, identifying common denominators, and simplifying fractions is crucial for success in algebra and beyond. So, let's embark on this journey together and master the art of simplifying rational expressions.

Understanding Rational Expressions

Before we dive into the specific problem of finding the difference between two rational expressions, it's essential to establish a solid understanding of what rational expressions are and the basic principles that govern their manipulation. Rational expressions, at their core, are fractions where both the numerator and the denominator are polynomials. A polynomial, in turn, is an expression consisting of variables and coefficients, combined using only the operations of addition, subtraction, and multiplication, with non-negative integer exponents. Examples of rational expressions abound in algebra, ranging from simple forms like x/y to more complex forms like (x^2 + 2x + 1) / (x^3 - 8). The key characteristic of rational expressions is that they represent a ratio of two polynomials, making them amenable to the rules and techniques of fraction manipulation.

The fundamental principle underlying the simplification of rational expressions is the concept of equivalent fractions. Just as we can simplify numerical fractions by dividing both the numerator and the denominator by a common factor, we can simplify rational expressions by dividing both the numerator and the denominator by a common polynomial factor. This process relies on the fact that multiplying or dividing both the numerator and the denominator of a fraction by the same non-zero quantity does not change the value of the fraction. This principle allows us to manipulate rational expressions while preserving their underlying mathematical value. In order to effectively simplify rational expressions, a strong foundation in factoring polynomials is essential. Factoring is the process of expressing a polynomial as a product of simpler polynomials. Common factoring techniques include factoring out the greatest common factor (GCF), factoring quadratic trinomials, and recognizing special patterns like the difference of squares or the sum/difference of cubes. By factoring the numerator and denominator of a rational expression, we can identify common factors that can be cancelled, leading to a simplified form of the expression.

The Problem: Finding the Difference

Let's tackle the problem at hand. We are tasked with finding the difference between two rational expressions:

3uu2−12u+36−18u2−12u+36\frac{3u}{u^2 - 12u + 36} - \frac{18}{u^2 - 12u + 36}

This problem presents a classic scenario in algebra, where we need to subtract two fractions with polynomial expressions. The expressions may seem intimidating initially, but we can simplify the problem by breaking it down into manageable steps. Our primary goal is to combine these two expressions into a single, simplified rational expression. To achieve this, we need to leverage our understanding of rational expressions and the techniques we discussed earlier.

The first crucial observation is that the two rational expressions share a common denominator. This is a significant advantage, as it allows us to directly subtract the numerators without the need to find a common denominator. When subtracting fractions with a common denominator, we simply subtract the numerators and keep the denominator the same. However, before we rush into subtraction, it is always prudent to examine the expressions for potential simplifications. In this case, we can see that the denominator is a quadratic trinomial, which might be factorable. Factoring the denominator can reveal common factors that we can cancel out, leading to a simpler expression. This step is essential for arriving at the simplest form of the result. Factoring the denominator not only helps in simplifying the expression but also provides valuable insights into the behavior of the expression, such as identifying any restrictions on the variable u (values that would make the denominator zero). By taking the time to factor and simplify, we can avoid potential pitfalls and arrive at the correct answer more efficiently.

Step 1: Factoring the Denominator

As we noted in the previous section, the denominator of both rational expressions is the quadratic trinomial u² - 12u + 36. Factoring this trinomial is the first crucial step in simplifying the expression. To factor a quadratic trinomial of the form ax² + bx + c, we look for two numbers that multiply to c and add up to b. In our case, a = 1, b = -12, and c = 36. We need to find two numbers that multiply to 36 and add up to -12.

By carefully considering the factors of 36, we can identify that -6 and -6 satisfy these conditions. (-6) * (-6) = 36 and (-6) + (-6) = -12. Therefore, we can factor the quadratic trinomial as follows:

u² - 12u + 36 = (u - 6)(u - 6)

This can also be written as:

u² - 12u + 36 = (u - 6)²

The factored form of the denominator reveals a perfect square trinomial. Recognizing perfect square trinomials is a valuable skill in algebra, as it allows for quicker factorization and simplification. Now that we have factored the denominator, we can rewrite the original expression with the factored form:

3u(u−6)2−18(u−6)2\frac{3u}{(u - 6)^2} - \frac{18}{(u - 6)^2}

This form is much more amenable to simplification. By factoring the denominator, we have laid the groundwork for the next step, which is subtracting the rational expressions. Factoring is not just a mechanical process; it is a powerful tool that unlocks the underlying structure of algebraic expressions. In this case, factoring the denominator has revealed the common factor (u - 6)², which will play a crucial role in the subsequent steps. This step highlights the importance of developing a strong factoring toolkit and applying it strategically when simplifying rational expressions.

Step 2: Subtracting the Expressions

Now that we have factored the denominator and rewritten the expression, we are ready to subtract the two rational expressions. Since they share a common denominator, this step is relatively straightforward. To subtract fractions with a common denominator, we simply subtract the numerators and keep the denominator the same. In our case, the common denominator is (u - 6)², and the numerators are 3u and 18. Therefore, the subtraction proceeds as follows:

3u(u−6)2−18(u−6)2=3u−18(u−6)2\frac{3u}{(u - 6)^2} - \frac{18}{(u - 6)^2} = \frac{3u - 18}{(u - 6)^2}

We have successfully subtracted the two rational expressions, resulting in a single rational expression with the combined numerator. However, our work is not yet complete. We must always look for opportunities to simplify the resulting expression further. The numerator, 3u - 18, appears to have a common factor, which we can factor out to potentially simplify the expression. Factoring the numerator is the next logical step in our quest to express the result in its simplest form. It's important to remember that simplification is not just about obtaining a correct answer; it's about expressing the answer in the most concise and elegant form possible. A simplified expression is easier to understand, manipulate, and interpret. This step underscores the iterative nature of simplification, where each step leads us closer to the final, simplified form. By systematically applying the techniques of factoring and simplification, we can confidently navigate the complexities of rational expressions.

Step 3: Simplifying the Result

As we observed in the previous step, the numerator of the resulting expression, 3u - 18, has a common factor. The greatest common factor (GCF) of 3u and 18 is 3. We can factor out the GCF from the numerator as follows:

3u - 18 = 3(u - 6)

Now we can rewrite the expression with the factored numerator:

3u−18(u−6)2=3(u−6)(u−6)2\frac{3u - 18}{(u - 6)^2} = \frac{3(u - 6)}{(u - 6)^2}

At this point, we can see a common factor in both the numerator and the denominator: (u - 6). This common factor can be cancelled, provided that u ≠ 6 (since division by zero is undefined). Cancelling the common factor, we get:

3(u−6)(u−6)2=3u−6\frac{3(u - 6)}{(u - 6)^2} = \frac{3}{u - 6}

This is the simplified form of the expression. We have successfully found the difference between the two rational expressions and expressed the result in its simplest form. The simplification process involved factoring, subtracting, and cancelling common factors. Each step was crucial in arriving at the final answer. This example illustrates the power of systematic problem-solving in mathematics. By breaking down a complex problem into smaller, manageable steps, we can tackle even the most challenging expressions. The final simplified form, 3/(u - 6), is not only more concise but also reveals important information about the expression's behavior, such as its domain (all real numbers except u = 6). This step highlights the importance of striving for simplicity and elegance in mathematical solutions.

Final Answer

Therefore, the difference between the two rational expressions, expressed in simplest form, is:

3u−6\frac{3}{u - 6}

This result represents the culmination of our step-by-step simplification process. We started with two seemingly complex rational expressions and, through the application of factoring, subtraction, and cancellation, arrived at a concise and elegant solution. This example demonstrates the power of algebraic manipulation and the importance of mastering the fundamental techniques of simplifying rational expressions.

In summary, simplifying rational expressions involves a series of key steps:

  1. Factoring: Factor the numerators and denominators of the expressions.
  2. Finding a Common Denominator: If the expressions do not have a common denominator, find the least common multiple (LCM) of the denominators and rewrite the expressions with the common denominator.
  3. Performing the Operation: Add or subtract the numerators, keeping the common denominator.
  4. Simplifying: Factor the resulting numerator and denominator and cancel any common factors.

By following these steps systematically, you can confidently simplify a wide range of rational expressions. The skills you develop in this process will be invaluable in more advanced mathematical studies, such as calculus and differential equations. Remember, practice is key to mastering these techniques. The more you work with rational expressions, the more comfortable and proficient you will become at simplifying them. So, embrace the challenge and embark on your journey to mathematical fluency.

By consistently applying these principles and practicing diligently, you can achieve mastery in simplifying rational expressions and unlock a deeper understanding of algebraic concepts.