Simplifying Rational Expressions A Step-by-Step Guide

Simplifying rational expressions can seem daunting at first, but with a systematic approach and a solid understanding of algebraic principles, it becomes a manageable task. This article will guide you through the process of simplifying the rational expression ${\frac{1}{2x^2 - 4x} - \frac{2}{x}}$ step by step, ensuring clarity and comprehension along the way. We will break down each stage, explaining the underlying concepts and techniques involved. This will not only help you solve this specific problem but also equip you with the skills to tackle similar challenges in the future. Let's embark on this mathematical journey together and master the art of simplifying rational expressions.

Understanding Rational Expressions

Before diving into the simplification process, it's essential to grasp the fundamental concept of rational expressions. A rational expression is essentially a fraction where the numerator and the denominator are polynomials. Just like with numerical fractions, we aim to reduce rational expressions to their simplest form. This involves identifying and canceling out common factors between the numerator and the denominator. The goal is to express the rational expression in a form where there are no more common factors to simplify. This not only makes the expression more concise but also facilitates further calculations and analysis. Understanding this basic principle is crucial for effectively simplifying any rational expression.

In our case, we are dealing with the expression ${\frac{1}{2x^2 - 4x} - \frac{2}{x}}$ which involves two rational expressions that need to be combined and simplified. The presence of variables in the denominators adds a layer of complexity, but by applying the principles of fraction manipulation and factorization, we can systematically simplify this expression. Let's move on to the first step in the simplification process: finding a common denominator.

Finding a Common Denominator

To combine the two rational expressions, ${\frac{1}{2x^2 - 4x}}$ and ${\frac{2}{x}}$ we need to find a common denominator. This is a crucial step in adding or subtracting any fractions, whether they involve numbers or algebraic expressions. The common denominator is the least common multiple (LCM) of the denominators of the fractions. In our case, the denominators are ${2x^2 - 4x}$ and ${x}$. To find the LCM, we first need to factor each denominator.

The first denominator, ${2x^2 - 4x}$, can be factored by taking out the greatest common factor, which is ${2x}$. This gives us ${2x(x - 2)}$. The second denominator, ${x}$, is already in its simplest form. Now, to find the LCM, we take the highest power of each unique factor present in the denominators. The factors are ${2}$, ${x}$, and ${x - 2}$. Therefore, the LCM is ${2x(x - 2)}$. This will be our common denominator.

Now that we have the common denominator, we need to rewrite each fraction with this new denominator. This involves multiplying the numerator and denominator of each fraction by the appropriate factors to achieve the common denominator. Let's proceed to the next step where we rewrite the fractions with the common denominator.

Rewriting Fractions with the Common Denominator

Now that we've identified the common denominator as ${2x(x - 2)}$, our next step is to rewrite each fraction with this denominator. The first fraction, ${\frac{1}{2x^2 - 4x}}$ already has the denominator ${2x^2 - 4x}$, which we factored as ${2x(x - 2)}$. So, this fraction already has the common denominator and doesn't need any modification.

The second fraction, ${\frac{2}{x}}$, needs to be adjusted. To get the common denominator of ${2x(x - 2)}$, we need to multiply both the numerator and the denominator by ${2(x - 2)}$. This gives us: ${\frac{2}{x} \times \frac{2(x - 2)}{2(x - 2)} = \frac{4(x - 2)}{2x(x - 2)}}$

Now, both fractions have the same denominator: ${2x(x - 2)}$. We can now rewrite our original expression as: ${\frac{1}{2x(x - 2)} - \frac{4(x - 2)}{2x(x - 2)}}$

With both fractions sharing the same denominator, we are now ready to combine them. This involves subtracting the numerators while keeping the common denominator. Let's move on to the next step of combining the fractions.

Combining the Fractions

Having rewritten both fractions with the common denominator ${2x(x - 2)}$, we can now combine them. This involves subtracting the numerators while keeping the common denominator. Our expression is: ${\frac{1}{2x(x - 2)} - \frac{4(x - 2)}{2x(x - 2)}}$

To combine the fractions, we subtract the second numerator from the first: ${\frac{1 - 4(x - 2)}{2x(x - 2)}}$

Now, we need to simplify the numerator by distributing the ${4}$ and combining like terms: ${\frac{1 - 4x + 8}{2x(x - 2)}}$ ${\frac{-4x + 9}{2x(x - 2)}}$

The numerator is now simplified to ${-4x + 9}$. Our expression now looks like this: ${\frac{-4x + 9}{2x(x - 2)}}$

This is a significant step towards simplifying the original expression. We have combined the two fractions into a single fraction. The next crucial step is to check if we can simplify further by factoring and canceling out any common factors between the numerator and the denominator. Let's move on to the final step of simplifying the resulting fraction.

Simplifying the Resulting Fraction

After combining the fractions, we arrived at the expression: ${\frac{-4x + 9}{2x(x - 2)}}$

Now, we need to check if this fraction can be simplified further. Simplification involves looking for common factors between the numerator and the denominator that can be canceled out. In this case, the numerator is ${-4x + 9}$, which is a linear expression. The denominator is ${2x(x - 2)}$, which is already factored.

We need to determine if there are any factors common to both the numerator and the denominator. The numerator, ${-4x + 9}$, cannot be factored easily, and it does not share any common factors with the terms in the denominator, which are ${2x}$ and ${(x - 2)}$.

Since there are no common factors to cancel out, the fraction is already in its simplest form. Therefore, the simplest form of the given expression is: ${\frac{-4x + 9}{2x(x - 2)}}$

This matches option A in the given choices. We have successfully simplified the rational expression by finding a common denominator, combining the fractions, and then checking for any further simplifications. Understanding each step and the underlying principles is key to mastering the simplification of rational expressions. By following this systematic approach, you can confidently tackle similar problems in the future.

Conclusion

In this article, we have walked through the process of simplifying the rational expression ${\frac{1}{2x^2 - 4x} - \frac{2}{x}}$ step by step. We started by understanding the concept of rational expressions and the importance of finding a common denominator. We then factored the denominators, identified the least common multiple, and rewrote each fraction with the common denominator. After combining the fractions, we simplified the numerator and checked for any common factors between the numerator and the denominator.

The final simplified form of the expression is: ${\frac{-4x + 9}{2x(x - 2)}}$

This corresponds to option A in the given choices. By breaking down the problem into manageable steps and understanding the underlying principles, we were able to simplify the rational expression effectively. Remember, the key to simplifying rational expressions lies in a systematic approach, careful factorization, and a thorough check for common factors. With practice and a solid understanding of these techniques, you can confidently tackle a wide range of simplification problems. This comprehensive guide should serve as a valuable resource for mastering the art of simplifying rational expressions.