Adding Rational Expressions A Step-by-Step Guide To Solving X/5 + 2/(3x)

In the realm of mathematics, rational expressions hold a significant position, particularly in algebra and calculus. These expressions, which are essentially fractions with polynomials in the numerator and denominator, require a unique set of rules and techniques for manipulation. One fundamental operation involving rational expressions is addition, which can sometimes appear daunting but becomes manageable with a clear understanding of the underlying principles. This comprehensive guide delves into the process of finding the sum of rational expressions, using the specific example of x/5 + 2/(3x) to illustrate the key steps and concepts involved.

Demystifying Rational Expressions: The Building Blocks

Before we embark on the journey of adding rational expressions, it's crucial to grasp what they are and how they differ from ordinary fractions. A rational expression is defined as a fraction where both the numerator and denominator are polynomials. Polynomials, in turn, are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of rational expressions include (x^2 + 1)/(x - 2), (3x)/(x^2 + 4), and, of course, the expressions we'll be working with: x/5 and 2/(3x). Understanding the structure of rational expressions is the foundation for performing operations on them.

The Golden Rule of Adding Fractions: Common Denominators

The cornerstone of adding any type of fraction, including rational expressions, lies in the concept of a common denominator. Just as you can't directly add apples and oranges, you can't directly add fractions with different denominators. The denominator represents the number of equal parts into which the whole is divided, and to combine fractions, we need to ensure they are divided into the same number of parts. This principle is paramount when dealing with rational expressions. To add x/5 and 2/(3x), we must first find a common denominator. This involves identifying a suitable expression that is divisible by both denominators, which in this case are 5 and 3x.

Finding the Least Common Denominator (LCD): The Key to Efficiency

While any common denominator will work, the most efficient approach involves finding the least common denominator (LCD). The LCD is the smallest expression that is a multiple of both denominators. In our example, the denominators are 5 and 3x. The LCD is the least common multiple (LCM) of 5 and 3x, which is simply their product: 15x. This is because 5 is a prime number, and 3x has prime factors of 3 and x, none of which are shared with 5 other than the implicit 1. Thus, the LCD, 15x, becomes our target denominator.

Transforming the Fractions: Equivalent Expressions

Once we've identified the LCD, the next step is to transform each rational expression into an equivalent expression with the LCD as the denominator. This is achieved by multiplying both the numerator and denominator of each fraction by a suitable factor. For the first fraction, x/5, we need to multiply the denominator 5 by 3x to obtain the LCD 15x. To maintain the fraction's value, we must also multiply the numerator x by 3x, resulting in (x * 3x) / (5 * 3x) = 3x^2 / 15x. Similarly, for the second fraction, 2/(3x), we need to multiply the denominator 3x by 5 to obtain the LCD 15x. Multiplying the numerator 2 by 5 gives us (2 * 5) / (3x * 5) = 10 / 15x. Now, both fractions have the same denominator, allowing us to proceed with addition.

Adding the Numerators: Combining Like Terms

With both fractions now sharing a common denominator of 15x, we can proceed to add them. The rule for adding fractions with the same denominator is simple: add the numerators and keep the denominator. Therefore, we have (3x^2 / 15x) + (10 / 15x) = (3x^2 + 10) / 15x. The numerator is the sum of the numerators from the original fractions, and the denominator remains the common denominator we found earlier. The next step is to examine the resulting expression to see if it can be simplified.

Simplifying the Result: The Final Touches

Simplifying rational expressions involves looking for common factors in the numerator and denominator that can be canceled out. In our case, the numerator is 3x^2 + 10, which is a quadratic expression. We need to check if it can be factored. The factors of 3x^2 would involve terms with x, and the factors of 10 are 1, 2, 5, and 10. However, there is no combination that would allow us to factor 3x^2 + 10 in a way that would cancel out with the denominator 15x. Since there are no common factors between the numerator 3x^2 + 10 and the denominator 15x, the expression is already in its simplest form. Therefore, the sum of the rational expressions x/5 + 2/(3x) is (3x^2 + 10) / 15x.

Checking the Options: Identifying the Correct Answer

Having arrived at the simplified sum (3x^2 + 10) / 15x, we can now compare it to the given options to identify the correct answer. The options provided are:

A. (x + 2) / (3x + 5) B. (3x^2 + 10) / 15x C. (3x^2 + 10) / (3x + 5) D. (x + 2) / 15x

By direct comparison, we can see that option B, (3x^2 + 10) / 15x, matches our calculated sum. Therefore, option B is the correct answer.

Common Mistakes to Avoid: A Word of Caution

Adding rational expressions can be tricky, and there are several common mistakes that students often make. One frequent error is attempting to add the numerators and denominators directly without finding a common denominator. This is incorrect and will lead to the wrong answer. Another mistake is failing to simplify the final result. Always check for common factors that can be canceled out. A further error lies in incorrectly identifying the LCD. It's crucial to find the least common denominator for simplicity. Finally, sign errors during the transformation and simplification steps are common, so careful attention to detail is essential.

Practice Makes Perfect: Reinforcing Your Skills

Mastering the addition of rational expressions, like any mathematical skill, requires practice. Work through a variety of examples, starting with simpler ones and gradually progressing to more complex problems. Pay close attention to each step, and don't hesitate to review the fundamental principles whenever needed. By consistently practicing, you'll develop the confidence and proficiency to tackle any rational expression addition problem that comes your way. Rational expressions are a cornerstone in algebra, calculus, and beyond. A solid grasp of manipulating them, particularly through addition, is an invaluable asset in mathematical studies.

What is the sum of the following rational expressions: x/5 + 2/(3x)?

Adding Rational Expressions A Step-by-Step Guide to Solving x/5 + 2/(3x)