Finding The Sum Of Rational Expressions A Step By Step Guide
When working with algebraic expressions, particularly rational expressions, it's crucial to understand how to simplify and combine them effectively. This article delves into the process of adding rational expressions, providing a step-by-step guide with detailed explanations and examples. In this comprehensive guide, we will address the question of finding the sum of rational expressions, specifically focusing on the expression: (3y)/(y^2 + 7y + 10) + 2/(y + 2). To effectively tackle this problem, we'll explore the fundamental principles of adding fractions with different denominators, a skill that's directly applicable to rational expressions. Understanding these principles is essential for anyone studying algebra or calculus, as these concepts form the foundation for more advanced mathematical operations.
Understanding Rational Expressions
Rational expressions are fractions where the numerator and denominator are polynomials. Just like with numerical fractions, adding rational expressions requires a common denominator. To add rational expressions, the first step is to ensure they share a common denominator. This often involves factoring the denominators and identifying the least common multiple (LCM). Once a common denominator is established, the numerators can be added together. For this particular problem, the given expression is (3y)/(y^2 + 7y + 10) + 2/(y + 2). We'll need to factor the quadratic expression in the denominator of the first term to find the common denominator. Factoring the denominator allows us to identify common factors and determine the least common denominator (LCD), which is crucial for simplifying the expression and combining the fractions effectively. Mastering these steps is essential for simplifying complex algebraic expressions and solving equations involving rational functions.
Factoring and Finding the Common Denominator
The first step in simplifying the expression (3y)/(y^2 + 7y + 10) + 2/(y + 2) is to factor the denominator of the first fraction. The quadratic expression y^2 + 7y + 10 can be factored into (y + 2)(y + 5). This factorization is a critical step because it reveals the common factors between the two denominators. Factoring is a fundamental skill in algebra, and it's essential for simplifying expressions and solving equations. By breaking down the quadratic into its linear factors, we can easily identify the terms needed to create a common denominator. Now, we can rewrite the original expression as (3y)/((y + 2)(y + 5)) + 2/(y + 2). The denominators are now (y + 2)(y + 5) and (y + 2). To find the least common denominator (LCD), we identify the unique factors and their highest powers. In this case, the LCD is (y + 2)(y + 5). The LCD is the smallest expression that both denominators divide into evenly, which is crucial for adding the fractions without changing their values. Understanding how to find the LCD is essential for adding and subtracting rational expressions effectively.
Adjusting the Fractions
Now that we have identified the least common denominator (LCD) as (y + 2)(y + 5), we need to rewrite each fraction with this common denominator. The first fraction, (3y)/((y + 2)(y + 5)), already has the LCD, so we don't need to change it. However, the second fraction, 2/(y + 2), needs to be adjusted. To get the LCD, we multiply both the numerator and denominator of the second fraction by the missing factor, which is (y + 5). This gives us (2(y + 5))/((y + 2)(y + 5)). Multiplying the numerator and denominator by the same factor ensures that we are only changing the form of the fraction, not its value. This is a fundamental principle in fraction manipulation and is crucial for adding rational expressions correctly. Now, we can rewrite the expression with the common denominator: (3y)/((y + 2)(y + 5)) + (2(y + 5))/((y + 2)(y + 5)). By ensuring both fractions have the same denominator, we can proceed to add the numerators, which is the next step in simplifying the expression.
Adding the Numerators
With both fractions now sharing the common denominator (y + 2)(y + 5), we can add the numerators. The expression becomes (3y + 2(y + 5))/((y + 2)(y + 5)). The next step is to simplify the numerator by distributing and combining like terms. Distributing the 2 in the numerator gives us 3y + 2y + 10. Combining the like terms 3y and 2y results in 5y + 10. So, the expression now looks like (5y + 10)/((y + 2)(y + 5)). Simplifying the numerator is a critical step in reducing the expression to its simplest form. By combining like terms, we make the expression easier to work with and reduce the chances of errors in subsequent steps. The simplified numerator allows us to potentially factor and further simplify the entire rational expression. This process highlights the importance of algebraic manipulation in simplifying complex expressions.
Simplifying the Result
After adding the numerators, we have the expression (5y + 10)/((y + 2)(y + 5)). Now, we can simplify further by factoring the numerator. The numerator 5y + 10 can be factored by taking out the common factor of 5, which gives us 5(y + 2). So, the expression becomes (5(y + 2))/((y + 2)(y + 5)). Factoring is a key step in simplifying rational expressions because it allows us to identify common factors in the numerator and denominator that can be canceled out. This cancellation is crucial for reducing the expression to its simplest form. Now, we can see that (y + 2) is a common factor in both the numerator and the denominator. Canceling this common factor, we get the simplified expression 5/(y + 5). This final simplification demonstrates the power of factoring and canceling common factors in reducing rational expressions to their most basic form. The result, 5/(y + 5), is the sum of the original expressions in its simplest form.
Final Answer
Therefore, the sum of the given rational expressions (3y)/(y^2 + 7y + 10) + 2/(y + 2) simplifies to 5/(y + 5). Looking at the provided options, the correct answer is C. 5/(y + 5). This process underscores the importance of mastering the steps involved in adding and simplifying rational expressions, including factoring, finding the least common denominator, adjusting fractions, adding numerators, and simplifying the result. Each step is crucial for arriving at the correct answer and demonstrating a strong understanding of algebraic principles. The ability to work with rational expressions is a fundamental skill in algebra and calculus, making it essential for students to practice and master these techniques. Understanding these concepts not only helps in solving problems but also builds a solid foundation for more advanced mathematical topics.
Conclusion
In conclusion, simplifying and adding rational expressions requires a systematic approach. First, factor the denominators to find the least common denominator (LCD). Second, adjust the fractions to have the LCD. Third, add the numerators. Finally, simplify the resulting expression by factoring and canceling common factors. Following these steps allows us to accurately find the sum of (3y)/(y^2 + 7y + 10) + 2/(y + 2), which is 5/(y + 5). Mastering these techniques is essential for success in algebra and calculus, as they form the building blocks for more complex mathematical operations. Remember, practice is key to developing proficiency in simplifying and adding rational expressions. By consistently applying these methods, you can confidently tackle a wide range of algebraic problems and enhance your mathematical skills. Understanding the underlying principles and practicing consistently will lead to mastery and a deeper appreciation for the elegance and power of mathematics.
