Simplifying Rational Expressions Adding Fractions With Polynomial Denominators

When dealing with the task of simplifying rational expressions, especially when adding fractions with polynomial denominators, it's essential to have a strong grasp of algebraic principles and techniques. This article delves into the step-by-step process of simplifying the given expression, $\frac{3}{2x+5} + \frac{5}{x-5}$, offering a detailed explanation that not only arrives at the solution but also enhances understanding of the underlying concepts. We will explore the crucial steps of finding a common denominator, combining fractions, and simplifying the resulting expression, making this a valuable resource for students and anyone looking to sharpen their algebra skills.

Finding the Least Common Denominator (LCD)

The cornerstone of adding fractions lies in identifying the Least Common Denominator (LCD). The Least Common Denominator is the smallest multiple that the denominators of the fractions share. In our expression, $\frac{3}{2x+5} + \frac{5}{x-5}$, we have two distinct denominators: $(2x+5)$ and $(x-5)$. Since these expressions do not share any common factors, the LCD is simply their product. Thus, the LCD is $(2x+5)(x-5)$. This step is crucial because it allows us to rewrite each fraction with a common base, making addition possible. Misunderstanding the LCD can lead to incorrect simplification, highlighting the importance of this initial step. To ensure a solid grasp, it's beneficial to practice identifying LCDs with various polynomial denominators, including those that require factoring before the LCD can be determined. By mastering this skill, you'll lay a strong foundation for tackling more complex rational expressions.

Rewriting Fractions with the LCD

Once the LCD is determined, the next crucial step involves rewriting each fraction with this common denominator. For the fraction $\frac{3}{2x+5}$, we need to multiply both the numerator and the denominator by $(x-5)$ to obtain the LCD in the denominator. This gives us $\frac{3(x-5)}{(2x+5)(x-5)}$. Similarly, for the fraction $\frac{5}{x-5}$, we multiply both the numerator and the denominator by $(2x+5)$, resulting in $\frac{5(2x+5)}{(2x+5)(x-5)}$. It's essential to remember that multiplying both the numerator and denominator by the same expression does not change the value of the fraction; it merely changes its form. This is a fundamental principle in fraction manipulation and is key to successfully adding and subtracting rational expressions. This process ensures that both fractions have a common denominator, allowing us to combine them in the next step. Attention to detail is paramount here, as any error in multiplication or distribution can lead to an incorrect final answer. Practice with various examples will solidify your understanding and improve your accuracy in rewriting fractions with a common denominator.

Combining the Fractions

With both fractions now sharing the common denominator $(2x+5)(x-5)$, we can proceed to combine the fractions. This involves adding the numerators while keeping the denominator the same. So, we have: $\frac{3(x-5) + 5(2x+5)}{(2x+5)(x-5)}$. This step is a direct application of the rule for adding fractions with common denominators: $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$. It's a straightforward process, but accuracy in the previous steps is crucial for a correct result. Before moving on, it's wise to double-check that the numerators have been correctly added and that the denominator remains unchanged. This careful review can prevent errors from propagating through the rest of the simplification process. The combined fraction sets the stage for the next step, which involves simplifying the numerator by distributing and combining like terms. Mastering this stage of combining fractions is pivotal in solving more complex algebraic problems involving rational expressions.

Simplifying the Numerator

The next critical step is simplifying the numerator of the combined fraction. This involves expanding any products and then combining like terms. Looking at our expression, $\frac{3(x-5) + 5(2x+5)}{(2x+5)(x-5)}$, we first distribute the 3 and the 5 across their respective parentheses: $3(x-5)$ becomes $3x - 15$, and $5(2x+5)$ becomes $10x + 25$. Now we have $\frac{3x - 15 + 10x + 25}{(2x+5)(x-5)}$. Next, we combine like terms in the numerator. We have two terms with $x$: $3x$ and $10x$, which combine to give $13x$. We also have two constant terms: $-15$ and $+25$, which combine to give $+10$. Therefore, the simplified numerator is $13x + 10$. This process highlights the importance of understanding the distributive property and how to correctly identify and combine like terms. Errors in this step can significantly impact the final result, so attention to detail is crucial. The ability to simplify numerators efficiently is a valuable skill in algebra, not just for rational expressions but for a wide range of algebraic manipulations.

Writing the Simplified Expression

Having simplified the numerator, we can now write the simplified expression. Our numerator has been reduced to $13x + 10$, and our denominator remains $(2x+5)(x-5)$. Therefore, the simplified fraction is $\frac{13x + 10}{(2x+5)(x-5)}$. This step represents the culmination of all the previous steps, bringing us to a concise form of the original expression. It's important to pause here and verify that the expression is indeed in its simplest form. This means checking that the numerator and denominator have no common factors that can be further simplified. In this case, $13x + 10$ and $(2x+5)(x-5)$ do not share any common factors, so our expression is fully simplified. The ability to accurately write the simplified expression is a testament to a solid understanding of algebraic principles and techniques. This final step not only provides the solution but also reinforces the importance of accuracy and attention to detail throughout the entire simplification process.

Therefore, the simplified form of the expression $\frac{3}{2x+5} + \frac{5}{x-5}$ is $\frac{13x+10}{(x-5)(2x+5)}$, which corresponds to option B.

Conclusion

In conclusion, simplifying rational expressions, particularly when adding fractions with polynomial denominators, involves a systematic approach. The key steps include finding the Least Common Denominator, rewriting fractions with the LCD, combining the fractions, and simplifying the numerator. Each step requires a clear understanding of algebraic principles and attention to detail. By mastering these steps, one can confidently tackle a wide range of algebraic problems involving rational expressions. The example we explored, $\frac{3}{2x+5} + \frac{5}{x-5}$, serves as a comprehensive illustration of this process. The ability to simplify complex expressions like this is a valuable skill in mathematics, with applications in various fields, including calculus, physics, and engineering. Continued practice and a solid foundation in algebra are crucial for success in this area. Remember, the journey of simplifying expressions is not just about arriving at the answer but also about developing a deeper understanding of mathematical concepts and techniques.