Simplifying Rational Expressions Step By Step Solution

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Understanding rational expressions and their simplification is crucial in algebra. In this article, we will delve into the process of simplifying a given rational expression, providing a step-by-step solution and offering insights into the underlying concepts. The problem at hand involves subtracting two rational expressions: 12x2−4x−2x\frac{1}{2x^2 - 4x} - \frac{2}{x}. To solve this, we need to find a common denominator, combine the fractions, and simplify the resulting expression. Mastering these steps will enhance your ability to tackle various algebraic problems involving rational expressions.

Understanding the Problem

The given expression is: 12x2−4x−2x\frac{1}{2x^2 - 4x} - \frac{2}{x}. This problem requires us to subtract two rational expressions. A rational expression is essentially a fraction where the numerator and the denominator are polynomials. To subtract these expressions, we need to find a common denominator, which is a crucial step in simplifying any fraction-related problem. The first denominator is 2x2−4x2x^2 - 4x, and the second is xx. Our goal is to manipulate these expressions to have the same denominator so that we can combine the numerators. This involves factoring the denominators and identifying the least common multiple (LCM), which will become our common denominator. By simplifying rational expressions, we aim to reduce them to their simplest form, making them easier to work with in further calculations or analyses. The process not only involves algebraic manipulation but also a solid understanding of factoring and fraction arithmetic.

Step-by-Step Solution

1. Factor the Denominators

The first step in simplifying the expression 12x2−4x−2x\frac{1}{2x^2 - 4x} - \frac{2}{x} is to factor the denominators. Factoring helps us identify common factors and determine the least common denominator (LCD). The first denominator, 2x2−4x2x^2 - 4x, can be factored by taking out the common factor of 2x2x. This gives us:

2x2−4x=2x(x−2)2x^2 - 4x = 2x(x - 2)

The second denominator, xx, is already in its simplest form and does not require further factoring. Factoring the denominators is a crucial step as it allows us to see the components we need for the common denominator. In this case, we see the factors 2x2x and (x−2)(x - 2) in the first denominator and xx in the second. Recognizing these factors is essential for the next step, which involves finding the least common denominator. Understanding factoring techniques is fundamental in simplifying rational expressions and is a skill that is widely applicable in algebra.

2. Find the Least Common Denominator (LCD)

Finding the least common denominator (LCD) is a critical step when adding or subtracting fractions, including rational expressions. The LCD is the smallest multiple that both denominators can divide into evenly. In our expression, 12x2−4x−2x\frac{1}{2x^2 - 4x} - \frac{2}{x}, we have already factored the first denominator as 2x(x−2)2x(x - 2) and the second denominator is xx. To find the LCD, we need to identify all unique factors present in the denominators and take the highest power of each factor.

  • The factors in the first denominator are 22, xx, and (x−2)(x - 2).
  • The factor in the second denominator is xx.

Thus, the LCD must include 22, xx, and (x−2)(x - 2). Therefore, the LCD is 2x(x−2)2x(x - 2).

Understanding how to find the LCD is essential because it allows us to rewrite the fractions with a common base, making it possible to combine them. This process is not only useful in algebra but also in various mathematical contexts where fraction manipulation is required. Correctly identifying the LCD ensures that we can accurately combine rational expressions without altering their values.

3. Rewrite the Fractions with the LCD

Now that we have found the least common denominator (LCD), which is 2x(x−2)2x(x - 2), the next step is to rewrite each fraction with this LCD. This involves multiplying the numerator and the denominator of each fraction by the necessary factors to achieve the common denominator. For the first fraction, 12x2−4x\frac{1}{2x^2 - 4x}, we already have the denominator factored as 2x(x−2)2x(x - 2), which matches our LCD. Therefore, we don't need to change this fraction.

For the second fraction, 2x\frac{2}{x}, we need to multiply both the numerator and the denominator by 2(x−2)2(x - 2) to get the LCD. This gives us:

2x×2(x−2)2(x−2)=4(x−2)2x(x−2)\frac{2}{x} \times \frac{2(x - 2)}{2(x - 2)} = \frac{4(x - 2)}{2x(x - 2)}

Now both fractions have the same denominator, 2x(x−2)2x(x - 2). Rewriting fractions with a common denominator is a fundamental step in adding and subtracting rational expressions. It ensures that we are combining like terms and maintaining the integrity of the expression. This step is crucial for simplifying complex algebraic expressions and is a skill that is widely used in various mathematical applications.

4. Combine the Numerators

After rewriting the fractions with the least common denominator (LCD), we can now combine the numerators. Our expression looks like this:

12x(x−2)−4(x−2)2x(x−2)\frac{1}{2x(x - 2)} - \frac{4(x - 2)}{2x(x - 2)}

Since the denominators are the same, we can subtract the numerators:

1−4(x−2)2x(x−2)\frac{1 - 4(x - 2)}{2x(x - 2)}

Now, we need to simplify the numerator by distributing the −4-4 across the terms inside the parentheses:

1−4x+82x(x−2)\frac{1 - 4x + 8}{2x(x - 2)}

Combining the constants in the numerator, we get:

9−4x2x(x−2)\frac{9 - 4x}{2x(x - 2)}

Combining numerators is a crucial step in simplifying rational expressions because it brings us closer to the simplest form of the expression. This process involves basic algebraic operations such as distribution and combining like terms, which are essential skills in mathematics. Ensuring that the numerators are correctly combined is vital for the accuracy of the final result.

5. Simplify the Result

After combining the numerators, we have the expression 9−4x2x(x−2)\frac{9 - 4x}{2x(x - 2)}. The final step is to simplify the result as much as possible. We check if there are any common factors in the numerator and the denominator that can be canceled out. In this case, the numerator is 9−4x9 - 4x, and the denominator is 2x(x−2)2x(x - 2).

There are no common factors between the numerator and the denominator that can be canceled. The expression is now in its simplest form.

So, the simplified expression is:

9−4x2x(x−2)\frac{9 - 4x}{2x(x - 2)}

This matches option A, which is −4x+92x(x−2)\frac{-4x + 9}{2x(x - 2)}.

Simplifying the result is a critical part of working with rational expressions. It ensures that the expression is in its most concise and understandable form. This step often involves factoring and canceling common factors, which requires a solid understanding of algebraic principles. The ability to simplify expressions is not only useful in academic settings but also in various practical applications where mathematical models need to be streamlined for efficiency.

Final Answer

The correct answer is A. −4x+92x(x−2)\frac{-4x + 9}{2x(x - 2)}.

Common Mistakes to Avoid

When simplifying rational expressions, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid errors and improve your accuracy. Here are some of the most frequent mistakes:

  1. Incorrect Factoring: Factoring is a crucial step in simplifying rational expressions. A common mistake is not factoring the denominators correctly or completely. For example, failing to factor 2x2−4x2x^2 - 4x as 2x(x−2)2x(x - 2) can lead to an incorrect LCD and subsequent errors. Always double-check your factoring to ensure it is accurate.

  2. Errors in Finding the LCD: The least common denominator (LCD) must include all unique factors from both denominators. A mistake here can lead to incorrect rewriting of fractions. Make sure to consider all factors and their highest powers when determining the LCD.

  3. Distributing Negatives Incorrectly: When subtracting rational expressions, it’s essential to distribute the negative sign correctly across the entire numerator of the fraction being subtracted. For instance, in our problem, subtracting 4(x−2)2x(x−2)\frac{4(x - 2)}{2x(x - 2)} requires distributing the −4-4 to both xx and −2-2, resulting in −4x+8-4x + 8. Failing to do this correctly can lead to significant errors.

  4. Incorrectly Combining Numerators: After finding the LCD and rewriting the fractions, students sometimes make mistakes when combining the numerators. This can involve errors in addition, subtraction, or combining like terms. Always double-check the arithmetic in this step.

  5. Forgetting to Simplify: The final step in simplifying rational expressions is to ensure the result is in its simplest form. This means looking for any common factors in the numerator and denominator that can be canceled out. Forgetting to simplify can result in an incomplete answer.

  6. Canceling Terms Instead of Factors: A common error is canceling terms that are not factors. For example, in the expression 9−4x2x(x−2)\frac{9 - 4x}{2x(x - 2)}, you cannot cancel the xx in the numerator with the xx in the denominator because 9−4x9 - 4x is a single term. Only common factors can be canceled.

  7. Not Checking for Excluded Values: Rational expressions may have excluded values, which are values of the variable that make the denominator zero. While not directly related to the simplification process, it’s important to identify these values to understand the domain of the expression. In our problem, xx cannot be 00 or 22.

By being mindful of these common mistakes, you can improve your accuracy and confidence in simplifying rational expressions.

Practice Problems

To reinforce your understanding of simplifying rational expressions, here are some practice problems. Working through these will help you develop the skills and confidence needed to tackle more complex problems.

  1. Simplify: 3x+1+4x−2\frac{3}{x + 1} + \frac{4}{x - 2}
  2. Simplify: 52x−1−2x+3\frac{5}{2x - 1} - \frac{2}{x + 3}
  3. Simplify: xx2−4+2x+2\frac{x}{x^2 - 4} + \frac{2}{x + 2}
  4. Simplify: x+1x2−2x−3−1x−3\frac{x + 1}{x^2 - 2x - 3} - \frac{1}{x - 3}
  5. Simplify: 2xx2+3x+2+xx+1\frac{2x}{x^2 + 3x + 2} + \frac{x}{x + 1}

For each problem, follow the steps we’ve discussed:

  • Factor the denominators.
  • Find the least common denominator (LCD).
  • Rewrite the fractions with the LCD.
  • Combine the numerators.
  • Simplify the result.

Working through these problems will give you practical experience in simplifying rational expressions and help you avoid common mistakes. Make sure to show your work and check your answers to ensure you understand each step of the process. Practice is key to mastering algebraic skills, and these problems provide an excellent opportunity to apply what you’ve learned.

Conclusion

In this article, we have walked through the process of simplifying the rational expression 12x2−4x−2x\frac{1}{2x^2 - 4x} - \frac{2}{x}. We began by factoring the denominators, which allowed us to identify the least common denominator (LCD). We then rewrote the fractions with the LCD, combined the numerators, and simplified the resulting expression. The correct answer was found to be −4x+92x(x−2)\frac{-4x + 9}{2x(x - 2)}, which corresponds to option A.

Simplifying rational expressions is a fundamental skill in algebra. It involves several key steps, including factoring, finding the LCD, rewriting fractions, combining numerators, and simplifying the final result. Understanding and mastering these steps is essential for success in algebra and related fields.

We also discussed common mistakes to avoid, such as incorrect factoring, errors in finding the LCD, improper distribution of negatives, and failing to simplify the final result. Being aware of these pitfalls can help you approach problems with greater confidence and accuracy.

Finally, we provided practice problems to give you an opportunity to apply what you’ve learned. Practice is crucial for solidifying your understanding and developing the skills necessary to tackle more complex problems.

By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying rational expressions. This skill will not only help you in your algebra studies but also in various other areas of mathematics and science where algebraic manipulation is required. Mastering these techniques will enhance your problem-solving abilities and provide a solid foundation for future mathematical endeavors.