Adding Rational Expressions Solve 3/(y^2-3y+2) + 5/(y^2-1)

Adding rational expressions often presents a challenge to students, but with a systematic approach, these problems can be solved effectively. This comprehensive guide will walk you through the process of adding the rational expressions ${\frac{3}{y^2-3y+2} + \frac{5}{y^2-1}}$, providing a clear, step-by-step explanation that enhances understanding and problem-solving skills. Let's dive into the detailed solution.

1. Understanding Rational Expressions

Before tackling the problem, it's crucial to understand what rational expressions are. A rational expression is essentially a fraction where the numerator and the denominator are polynomials. Just like adding regular fractions, adding rational expressions requires a common denominator. The key is to find the least common denominator (LCD) to simplify the process. The LCD is the least common multiple (LCM) of the denominators.

To illustrate, consider the expressions in our problem: ${\frac{3}{y^2-3y+2}}$ and ${\frac{5}{y^2-1}}$. The denominators are polynomials: y² - 3y + 2 and y² - 1. To add these expressions, we must first find the LCD of these denominators. This involves factoring each polynomial and identifying the common and unique factors. Factoring polynomials is a fundamental skill in algebra, and mastering it is essential for success in handling rational expressions. Understanding the structure of rational expressions and the importance of the LCD sets the stage for a smooth solution process. This groundwork ensures that you approach the problem with a clear strategy, making the subsequent steps more manageable and less prone to error. The concept of LCD is not just limited to this particular problem but is a universal principle in adding any kind of fractions, be it numerical or algebraic. By grasping this fundamental concept, you're equipping yourself with a powerful tool that can be applied in various mathematical contexts. Additionally, recognizing the characteristics of polynomials and how they behave within rational expressions provides a deeper insight into algebraic manipulations. This initial understanding forms the backbone of your approach, enabling you to navigate through the complexities of the problem with confidence and precision. By investing time in understanding the basics, you're setting a strong foundation for more advanced topics in algebra and beyond. Therefore, ensure you're comfortable with the definitions and principles discussed here before moving forward.

2. Factoring the Denominators

The next critical step in adding rational expressions is factoring the denominators. Factoring allows us to identify the individual components that make up each denominator, which is essential for finding the least common denominator (LCD). In our problem, we have two denominators: y² - 3y + 2 and y² - 1. Let's factor each one:

Factoring ${y^2 - 3y + 2}$

This is a quadratic expression. We need to find two numbers that multiply to 2 and add to -3. Those numbers are -1 and -2. Therefore, we can factor the quadratic expression as:

${y^2 - 3y + 2 = (y - 1)(y - 2)}$

Factoring ${y^2 - 1}$

This expression is a difference of squares, which follows the pattern ${a^2 - b^2 = (a + b)(a - b)}$. In this case, a = y and b = 1. So, we can factor it as:

${y^2 - 1 = (y + 1)(y - 1)}$

Now that we have factored both denominators, we can clearly see their components. The first denominator, y² - 3y + 2, factors into (y - 1)(y - 2), while the second denominator, y² - 1, factors into (y + 1)(y - 1). These factored forms are crucial for identifying the LCD. Factoring is a pivotal step because it simplifies the process of finding the LCD. Without factoring, it would be challenging to determine the common and unique factors needed to construct the LCD. Think of factoring as breaking down a complex problem into smaller, more manageable parts. By identifying the factors, you can easily see the building blocks of each denominator and how they relate to each other. This clarity is invaluable when determining the LCD and moving forward with the addition process. Furthermore, mastering factoring techniques is not just beneficial for this specific type of problem; it's a fundamental skill that's applicable across various areas of mathematics, including algebra, calculus, and beyond. Therefore, investing time in perfecting your factoring skills is a worthwhile endeavor that will pay dividends in your mathematical journey. Make sure you practice different types of factoring problems, such as factoring quadratics, differences of squares, and other common patterns. This will not only improve your speed and accuracy but also enhance your overall understanding of algebraic manipulations.

3. Finding the Least Common Denominator (LCD)

After factoring the denominators, the next essential step is to find the least common denominator (LCD). The LCD is the smallest expression that is a multiple of both denominators. It's the key to adding rational expressions because it allows us to combine the fractions into a single expression. In our problem, the factored denominators are (y - 1)(y - 2) and (y + 1)(y - 1).

To find the LCD, we need to identify all the unique factors present in the denominators and include each factor with the highest power it appears in any of the denominators. Looking at our factored denominators, we have the following factors:

• (y - 1)
• (y - 2)
• (y + 1)

The factor (y - 1) appears in both denominators, so we include it once. The factors (y - 2) and (y + 1) appear only once in their respective denominators, so we include them as well. Therefore, the LCD is the product of these unique factors:

${LCD = (y - 1)(y - 2)(y + 1)}$

The LCD is crucial because it serves as the common ground for both fractions, allowing us to add them together. Without a common denominator, we cannot directly add the numerators. The LCD acts as a bridge, connecting the two rational expressions and enabling us to perform the addition. Understanding how to find the LCD is a fundamental skill in working with rational expressions. It's not just a mechanical process; it's about understanding the relationships between the denominators and identifying the smallest expression that accommodates both. This concept is analogous to finding the least common multiple (LCM) when adding numerical fractions. The LCD ensures that when we rewrite the fractions with a common denominator, we are using the smallest possible expression, which simplifies the subsequent steps. This simplification is vital because it reduces the complexity of the problem and makes it easier to manage. Moreover, correctly identifying the LCD minimizes the chances of errors in the later stages of the solution. By carefully considering all the factors and their highest powers, you can confidently determine the LCD and proceed with the addition process. This step is a cornerstone of adding rational expressions, and mastering it will significantly improve your problem-solving abilities in algebra and beyond.

4. Rewriting the Fractions with the LCD

Once we have the LCD, the next step is to rewrite each fraction with the LCD as its denominator. This involves multiplying the numerator and denominator of each fraction by the factors needed to obtain the LCD. Recall our original fractions and the LCD:

• Fractions: ${\frac{3}{y^2-3y+2}}$ and ${\frac{5}{y^2-1}}$
• Factored Denominators: (y - 1)(y - 2) and (y + 1)(y - 1)
• LCD: (y - 1)(y - 2)(y + 1)

Now, let's rewrite each fraction:

Rewriting ${\frac{3}{y^2-3y+2}}$

The denominator y² - 3y + 2 is equivalent to (y - 1)(y - 2). To get the LCD, (y - 1)(y - 2)(y + 1), we need to multiply the denominator by (y + 1). To keep the fraction equivalent, we must also multiply the numerator by (y + 1):

${\frac{3}{(y - 1)(y - 2)} \times \frac{(y + 1)}{(y + 1)} = \frac{3(y + 1)}{(y - 1)(y - 2)(y + 1)}}$

Rewriting ${\frac{5}{y^2-1}}$

The denominator y² - 1 is equivalent to (y + 1)(y - 1). To get the LCD, (y - 1)(y - 2)(y + 1), we need to multiply the denominator by (y - 2). Again, we must multiply the numerator by the same factor:

${\frac{5}{(y + 1)(y - 1)} \times \frac{(y - 2)}{(y - 2)} = \frac{5(y - 2)}{(y - 1)(y - 2)(y + 1)}}$

Now both fractions have the LCD as their denominator. Rewriting fractions with a common denominator is a critical step because it sets the stage for the actual addition. By ensuring that both fractions have the same denominator, we can directly add their numerators, which is the core of fraction addition. This process involves careful multiplication of both the numerator and the denominator by the appropriate factors. It's crucial to ensure that you're multiplying by the correct factors to achieve the LCD without changing the value of the fraction. The principle behind this step is the fundamental property of fractions: multiplying both the numerator and the denominator by the same non-zero factor does not change the fraction's value. Think of it as scaling the fraction up or down while maintaining its proportion. This step requires attention to detail and a clear understanding of how the factors relate to each other. By accurately rewriting the fractions, you ensure that the subsequent addition step is straightforward and leads to the correct result. This methodical approach minimizes the chances of errors and allows you to confidently proceed with the final steps of the problem. Remember, the goal is to create a common ground for the two fractions, and this step precisely achieves that by aligning their denominators.

5. Adding the Numerators

With both fractions now having the same denominator, we can proceed to add the numerators. This involves combining the numerators over the common denominator. Our rewritten fractions are:

${\frac{3(y + 1)}{(y - 1)(y - 2)(y + 1)} \quad \text{and} \quad \frac{5(y - 2)}{(y - 1)(y - 2)(y + 1)}}$

Adding the numerators gives us:

${\frac{3(y + 1) + 5(y - 2)}{(y - 1)(y - 2)(y + 1)}}$

Now, we need to simplify the numerator by distributing and combining like terms:

${\frac{3y + 3 + 5y - 10}{(y - 1)(y - 2)(y + 1)}}$

Combining like terms in the numerator, we get:

${\frac{8y - 7}{(y - 1)(y - 2)(y + 1)}}$

Adding the numerators is a direct application of the principle of fraction addition. Once the fractions share a common denominator, adding them becomes a matter of combining the numerators while keeping the denominator the same. This step is where the hard work of finding the LCD and rewriting the fractions pays off. The process involves carefully adding the expressions in the numerators, which may include distributing, combining like terms, and simplifying the resulting expression. The key to success in this step is attention to detail. Make sure you correctly distribute any coefficients, combine like terms accurately, and watch out for sign errors. A small mistake in the numerator can propagate through the rest of the solution, leading to an incorrect answer. Therefore, take your time and double-check your work at this stage. The simplified numerator represents the combined expression from the original fractions, but it's not the final answer yet. We still need to ensure that the rational expression is in its simplest form. This often involves checking for further factoring opportunities or any common factors between the numerator and the denominator that can be canceled out. Adding the numerators is a crucial step, but it's just one part of the overall process. The final solution will depend on how well you simplify the resulting expression. Therefore, after adding the numerators, always take a moment to review and ensure that you've combined the terms correctly and are ready to move on to the simplification stage.

6. Simplifying the Result

After adding the numerators, the final step is to simplify the result. This involves checking if the numerator and denominator have any common factors that can be canceled out. Our expression is:

${\frac{8y - 7}{(y - 1)(y - 2)(y + 1)}}$

First, we look at the numerator, 8y - 7. There are no common factors that can be factored out of this expression. Next, we examine the denominator, (y - 1)(y - 2)(y + 1). We need to check if any of these factors can be canceled with the numerator.

In this case, 8y - 7 does not share any common factors with (y - 1), (y - 2), or (y + 1). Therefore, the expression is already in its simplest form. Thus, the final answer is:

${\frac{8y - 7}{(y - 1)(y - 2)(y + 1)}}$

Simplifying the result is a crucial step in any problem involving rational expressions. It ensures that your answer is in its most concise and understandable form. The process involves looking for common factors between the numerator and the denominator that can be canceled out. This is similar to reducing a numerical fraction to its lowest terms. The goal is to eliminate any redundant factors, leaving the expression in its simplest state. Simplifying often requires factoring both the numerator and the denominator to identify common factors. However, in some cases, like the one we have here, the numerator may not be factorable, or it may not share any factors with the denominator. If no common factors are found, it means the expression is already in its simplest form. Simplification is not just about making the expression look cleaner; it's also about ensuring the accuracy and clarity of the solution. A simplified expression is easier to work with in subsequent calculations and provides a more transparent representation of the mathematical relationship. Furthermore, simplifying the result is often a requirement in mathematical contexts, such as exams or problem sets. An unsimplified answer, even if mathematically correct, may not receive full credit. Therefore, it's essential to make simplification a standard part of your problem-solving routine. After performing any operation on rational expressions, always take the time to check for common factors and simplify the result. This will not only improve the presentation of your answer but also enhance your understanding of the underlying mathematical principles.

Conclusion

Adding rational expressions involves a series of steps, each building upon the previous one. From factoring the denominators and finding the LCD to rewriting the fractions and simplifying the result, each step is crucial for arriving at the correct answer. By following this systematic approach, you can confidently tackle similar problems and enhance your understanding of algebraic manipulations. Remember, practice is key to mastering these concepts. The ability to add rational expressions effectively is a fundamental skill in algebra and is essential for more advanced mathematical topics. By breaking down the problem into manageable steps and practicing consistently, you can build a strong foundation in this area.