Simplifying Rational Expressions A Comprehensive Guide To (2x)/(x^2-3x+2) + (2x)/(x-1) - X/(x-2)

In the realm of mathematics, particularly in algebra, simplifying rational expressions is a fundamental skill. These expressions, which involve fractions with polynomials in the numerator and denominator, often appear complex at first glance. However, with a systematic approach and a solid understanding of algebraic principles, they can be simplified into more manageable forms. This article delves into the process of simplifying the rational expression ${\frac{2x}{x^2-3x+2} + \frac{2x}{x-1} - \frac{x}{x-2}}$, providing a step-by-step guide and shedding light on the underlying concepts. Mastering the art of simplifying such expressions is not only crucial for academic success but also for various applications in engineering, physics, and computer science. This comprehensive guide aims to equip you with the knowledge and techniques necessary to confidently tackle similar problems. We will begin by dissecting the expression, identifying key components, and then systematically applying algebraic manipulations to arrive at the simplified form. The journey will involve factoring quadratic expressions, finding common denominators, combining fractions, and potentially simplifying the resulting expression further. By the end of this article, you will have a clear understanding of how to approach and solve such problems, enhancing your algebraic proficiency and problem-solving skills. Remember, practice is key to mastering these concepts, so work through the examples carefully and try similar problems on your own.

Understanding the Expression

Before diving into the simplification process, let's first understand the expression ${\frac{2x}{x^2-3x+2} + \frac{2x}{x-1} - \frac{x}{x-2}}$. This expression is a combination of three rational expressions, each involving polynomials. The first term, ${\frac{2x}{x^2-3x+2}}$, has a quadratic polynomial ${x^2-3x+2}$ in the denominator. The second term, ${\frac{2x}{x-1}}$, has a linear polynomial ${x-1}$ in the denominator, and the third term, ${\frac{x}{x-2}}$, also has a linear polynomial ${x-2}$ in the denominator. To simplify this expression, our primary goal is to combine these three fractions into a single, simplified fraction. This involves finding a common denominator, which is the least common multiple (LCM) of the denominators of the individual fractions. Factoring the denominators is a crucial step in this process, as it allows us to identify the common factors and determine the LCM efficiently. In this case, the quadratic denominator ${x^2-3x+2}$ can be factored into ${(x-1)(x-2)}$. This factorization reveals that the denominators share common factors, which will help us in finding the common denominator. Once we have the common denominator, we can rewrite each fraction with this denominator, add or subtract the numerators, and then simplify the resulting expression if possible. This may involve combining like terms, factoring the numerator, and canceling common factors between the numerator and denominator. The final simplified expression should be in its lowest terms, meaning that there are no common factors between the numerator and denominator. Understanding the structure of the expression and the steps involved in simplification is crucial for success in algebra and related fields.

Step-by-Step Simplification

Let's embark on the step-by-step simplification of the expression ${\frac{2x}{x^2-3x+2} + \frac{2x}{x-1} - \frac{x}{x-2}}$. The first and foremost step is to factor the quadratic denominator ${x^2-3x+2}$. We need to find two numbers that multiply to 2 and add up to -3. These numbers are -1 and -2. Therefore, we can factor the quadratic as ${(x-1)(x-2)}$. Now, the expression becomes ${\frac{2x}{(x-1)(x-2)} + \frac{2x}{x-1} - \frac{x}{x-2}}$. Next, we need to find the common denominator for the three fractions. The denominators are ${(x-1)(x-2)}$, ${(x-1)}$, and ${(x-2)}$. The least common multiple (LCM) of these denominators is ${(x-1)(x-2)}$. This is because it includes all the factors present in each denominator. Now, we rewrite each fraction with the common denominator ${(x-1)(x-2)}$. The first fraction already has this denominator, so it remains unchanged. For the second fraction, we multiply both the numerator and denominator by ${(x-2)}$, resulting in ${\frac{2x(x-2)}{(x-1)(x-2)}}$. For the third fraction, we multiply both the numerator and denominator by ${(x-1)}$, resulting in ${\frac{x(x-1)}{(x-1)(x-2)}}$. Now, the expression becomes ${\frac{2x}{(x-1)(x-2)} + \frac{2x(x-2)}{(x-1)(x-2)} - \frac{x(x-1)}{(x-1)(x-2)}}$. With a common denominator, we can combine the numerators: ${\frac{2x + 2x(x-2) - x(x-1)}{(x-1)(x-2)}}$. Now, we simplify the numerator by expanding and combining like terms.

Simplifying the Numerator

Continuing our step-by-step simplification, we now focus on simplifying the numerator of the expression. We have the numerator as ${2x + 2x(x-2) - x(x-1)}$. First, we distribute the terms: ${2x + 2x^2 - 4x - x^2 + x}$. Next, we combine like terms: ${(2x^2 - x^2) + (2x - 4x + x)}$, which simplifies to ${x^2 - x}$. Now, the expression becomes ${\frac{x^2 - x}{(x-1)(x-2)}}$. We can further simplify the numerator by factoring out a common factor of ${x}$: ${x(x-1)}$. So, the expression now looks like ${\frac{x(x-1)}{(x-1)(x-2)}}$. Notice that we have a common factor of ${(x-1)}$ in both the numerator and the denominator. We can cancel this common factor, provided that ${x \neq 1}$ (because division by zero is undefined). After canceling the common factor, we are left with ${\frac{x}{x-2}}$. This is the simplified form of the original expression. It's crucial to remember the restriction ${x \neq 1}$ because the original expression was undefined at ${x=1}$. This restriction ensures that we have not altered the domain of the expression during the simplification process. The simplified expression ${\frac{x}{x-2}}$ is a more compact and manageable form of the original expression, making it easier to work with in further calculations or applications. This step-by-step process highlights the importance of algebraic manipulation skills, such as factoring, distributing, combining like terms, and canceling common factors, in simplifying rational expressions.

Final Simplified Expression and Restrictions

After meticulously working through the steps, we have arrived at the final simplified expression: ${\frac{x}{x-2}}$. This expression is the most compact and simplified form of the original expression, ${\frac{2x}{x^2-3x+2} + \frac{2x}{x-1} - \frac{x}{x-2}}$. However, it's crucial to remember the restrictions on the variable ${x}$. These restrictions arise from the original expression's denominators. We factored the quadratic denominator ${x^2-3x+2}$ into ${(x-1)(x-2)}$. The other denominators were ${(x-1)}$ and ${(x-2)}$. Denominators cannot be equal to zero, as division by zero is undefined. Therefore, we must identify the values of ${x}$ that would make any of the original denominators zero. Setting each factor equal to zero, we have:

• ${x-1 = 0}$, which gives ${x = 1}$
• ${x-2 = 0}$, which gives ${x = 2}$

Thus, the original expression is undefined when ${x = 1}$ or ${x = 2}$. These values are the restrictions on the variable ${x}$. We must exclude these values from the domain of the expression. Even though we canceled the factor ${(x-1)}$ during the simplification process, the restriction ${x \neq 1}$ still applies because it was a restriction on the original expression. Therefore, the final simplified expression is ${\frac{x}{x-2}}$, with the restrictions ${x \neq 1}$ and ${x \neq 2}$. This means that the simplified expression is equivalent to the original expression for all values of ${x}$ except for ${x = 1}$ and ${x = 2}$. Including the restrictions is an essential part of the simplification process, as it ensures that the simplified expression is mathematically equivalent to the original expression over its entire domain.

Conclusion

In conclusion, simplifying rational expressions is a crucial skill in algebra and beyond. In this article, we have meticulously simplified the expression ${\frac{2x}{x^2-3x+2} + \frac{2x}{x-1} - \frac{x}{x-2}}$, step by step. The process involved factoring the quadratic denominator, finding a common denominator, combining the fractions, simplifying the numerator, and canceling common factors. The final simplified expression is ${\frac{x}{x-2}}$. However, we also emphasized the importance of identifying and stating the restrictions on the variable ${x}$. In this case, the restrictions are ${x \neq 1}$ and ${x \neq 2}$, which arise from the original expression's denominators. These restrictions ensure that the simplified expression is mathematically equivalent to the original expression over its entire domain. Throughout this process, we highlighted key algebraic techniques, such as factoring, distributing, combining like terms, and canceling common factors. Mastering these techniques is essential for simplifying rational expressions and tackling more complex algebraic problems. Furthermore, understanding the concept of restrictions and their importance in maintaining the equivalence of expressions is crucial for mathematical accuracy and rigor. Simplifying rational expressions is not just an exercise in algebraic manipulation; it's a process that enhances problem-solving skills, logical reasoning, and mathematical precision. By following a systematic approach and paying attention to detail, you can confidently simplify rational expressions and apply these skills to various mathematical and real-world problems. Remember, practice is key to mastering these concepts, so continue to work through examples and challenge yourself with increasingly complex problems. This will solidify your understanding and build your confidence in algebraic manipulation.