Simplifying Rational Expressions Understanding The Difference

This article aims to clarify the process of simplifying and subtracting rational expressions, specifically focusing on the given example. We will break down each step, highlighting key concepts in algebra, such as factoring quadratic expressions, finding common denominators, and simplifying resulting expressions. Our journey will involve a detailed examination of the initial problem and the subsequent steps taken to arrive at the final simplified form. By the end of this guide, you will have a solid understanding of how to tackle similar problems involving rational expressions.

Deconstructing the Initial Problem

Our initial challenge lies in understanding the difference between two rational expressions: x / (x^2 - 2x - 15) and 4 / (x^2 + 2x - 35). To effectively tackle this, we must first recognize that we are dealing with fractions where the numerators and denominators are polynomials. The core strategy here is akin to subtracting regular fractions: we need a common denominator. However, before we jump into finding the common denominator, it is often beneficial to factor the denominators to simplify the process.

The first denominator, x^2 - 2x - 15, is a quadratic expression. Factoring this quadratic expression involves finding two numbers that multiply to -15 and add to -2. These numbers are -5 and +3. Therefore, we can factor the denominator as (x - 5)(x + 3). This factorization is crucial as it allows us to rewrite the first rational expression in a more manageable form. By breaking down the quadratic into its linear factors, we gain a clearer picture of the expression's structure and how it might interact with other expressions. Understanding the roots of the quadratic also provides insight into where the expression might be undefined, which is a critical consideration in advanced algebraic manipulations.

Similarly, the second denominator, x^2 + 2x - 35, is another quadratic expression that needs factoring. This time, we are looking for two numbers that multiply to -35 and add to +2. These numbers are +7 and -5. Hence, we can factor this denominator as (x + 7)(x - 5). This factorization is equally important as it reveals the underlying factors of the second rational expression. Recognizing the common factor of (x - 5) between the two denominators is a significant observation, as it will play a key role in determining the common denominator for the subtraction.

By factoring both denominators, we transform the original problem into a more transparent form. The original expression, which may have seemed daunting at first, now reveals its underlying structure through these factorizations. This step is fundamental in simplifying rational expressions and is a testament to the power of factoring in algebraic manipulations. Factoring not only simplifies the expressions but also provides insights into the behavior and characteristics of the functions represented by these expressions.

Finding the Common Denominator

With the denominators factored as (x - 5)(x + 3) and (x + 7)(x - 5), the next crucial step is to identify the common denominator. The common denominator is the least common multiple (LCM) of the two denominators. To find the LCM, we consider all the unique factors present in the denominators and their highest powers. In this case, we have three unique factors: (x - 5), (x + 3), and (x + 7). The common denominator, therefore, is the product of these factors: (x - 5)(x + 3)(x + 7).

Understanding the concept of a common denominator is paramount when dealing with addition and subtraction of fractions, including rational expressions. It provides a uniform base for combining the fractions, ensuring that we are operating on like terms. The common denominator acts as a bridge, allowing us to perform arithmetic operations across fractions with different denominators. Without a common denominator, the numerators cannot be directly combined, and the operation would be mathematically incorrect. The process of finding the LCM ensures that the common denominator is the smallest possible expression that is divisible by both original denominators, thus simplifying subsequent calculations.

Now that we have the common denominator, we need to rewrite each rational expression with this new denominator. This involves multiplying the numerator and denominator of each fraction by the factors that are missing from its original denominator. For the first fraction, x / (x - 5)(x + 3), we need to multiply both the numerator and denominator by (x + 7). This gives us x(x + 7) / (x - 5)(x + 3)(x + 7). This step is crucial in maintaining the value of the fraction while adapting it to the common denominator. Multiplying both the numerator and denominator by the same factor is equivalent to multiplying by 1, which does not change the fraction's value but alters its representation.

Similarly, for the second fraction, 4 / (x + 7)(x - 5), we need to multiply both the numerator and denominator by (x + 3). This results in 4(x + 3) / (x - 5)(x + 3)(x + 7). Again, this process ensures that we are working with equivalent fractions that share a common denominator. By carefully considering the missing factors and multiplying both the numerator and denominator accordingly, we prepare the expressions for the final subtraction step. The resulting fractions now have the same denominator, allowing us to combine their numerators and simplify the expression further.

Performing the Subtraction

With both rational expressions now sharing the common denominator (x - 5)(x + 3)(x + 7), we can proceed with the subtraction. This involves subtracting the numerators while keeping the common denominator. The expression becomes:

[x(x + 7) - 4(x + 3)] / (x - 5)(x + 3)(x + 7)

This step is a direct application of the rules of fraction subtraction. Once the fractions have a common denominator, the subtraction is performed on the numerators, effectively combining the expressions into a single fraction. The common denominator acts as a constant base, while the numerators are subjected to the subtraction operation. This step streamlines the expression and sets the stage for further simplification.

Next, we need to simplify the numerator by expanding the terms and combining like terms. Expanding x(x + 7) gives us x^2 + 7x, and expanding -4(x + 3) gives us -4x - 12. Therefore, the numerator becomes:

x^2 + 7x - 4x - 12

Simplifying this expression by combining like terms, we get:

x^2 + 3x - 12

This simplification is a crucial step in reducing the complexity of the expression. By expanding and combining like terms, we distill the numerator into a more concise and manageable form. This process not only makes the expression easier to work with but also reveals potential patterns or simplifications that might not have been apparent in the expanded form. The simplified numerator now provides a clearer picture of the expression's behavior and its relationship to the denominator.

The Final Simplified Expression

After performing the subtraction and simplifying the numerator, we arrive at the expression:

(x^2 + 3x - 12) / (x - 5)(x + 3)(x + 7)

This expression represents the simplified form of the original subtraction problem. It is the result of carefully factoring the denominators, finding a common denominator, rewriting the fractions, subtracting the numerators, and simplifying the resulting expression. This final form allows for a clearer understanding of the relationship between the variables and the overall behavior of the expression. While this expression might not be further simplifiable through factoring (since the quadratic in the numerator does not factor neatly), it is in its most reduced form in terms of algebraic operations.

Now, let's examine the steps presented in the original problem and compare them to our derived solution. The steps provided in the problem lead to two different expressions:

1. (x^2 + 3x + 12) / (x - 3)(x - 5)(x + 7)
2. (x^2 + 3x - 12) / (x + 3)(x - 5)(x + 7)

Our derived solution matches the second expression. However, it's important to note a discrepancy in the provided steps. The step x(x + 3 - 12) / (x + 3)(x - 5)(x + 7) appears to be an error, as it incorrectly simplifies the numerator. This highlights the importance of carefully reviewing each step in algebraic manipulations to ensure accuracy. Errors in simplification can lead to incorrect results, and it's crucial to double-check each step to maintain the integrity of the solution.

The correct simplification process, as we have demonstrated, involves expanding the terms in the numerator and combining like terms. This ensures that the expression is simplified according to the rules of algebra and that the final result accurately represents the original problem. By understanding the underlying principles and meticulously applying the correct steps, we can confidently navigate the complexities of rational expressions and arrive at the correct solution.

Conclusion

In conclusion, simplifying rational expressions requires a systematic approach involving factoring, finding common denominators, and careful algebraic manipulation. By breaking down the problem into smaller, manageable steps, we can effectively tackle even complex expressions. The key takeaways from this exercise include the importance of accurate factoring, the necessity of finding a common denominator for subtraction, and the careful simplification of the resulting expression. The discrepancy in the provided steps serves as a reminder to always double-check each step in the simplification process to avoid errors. Through practice and a solid understanding of algebraic principles, you can confidently navigate the world of rational expressions and achieve accurate results.