Solving Rational Expressions A Step By Step Guide To (6-1)/(2x^2) - 3/x

In the realm of mathematics, particularly algebra, rational expressions play a crucial role. These expressions, which are essentially fractions with polynomials in the numerator and denominator, require specific techniques for operations like subtraction. In this comprehensive guide, we will dissect the problem ${\frac{6-1}{2x^2} - \frac{3}{x}}$, providing a step-by-step solution and explaining the underlying concepts. Understanding how to subtract rational expressions is fundamental for more advanced algebraic manipulations and problem-solving. Let's embark on this mathematical journey to master the art of subtracting these expressions.

Demystifying Rational Expressions

To effectively subtract rational expressions, it’s crucial to first understand what they are and the basic principles governing their manipulation. A rational expression is a fraction where both the numerator and the denominator are polynomials. Polynomials, in turn, are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include ${x^2 + 3x - 2}$, ${5x^3 - 1}$, and even simple terms like ${7}$ or ${x}$. Rational expressions appear frequently in algebra and calculus, making their understanding paramount.

The key to working with rational expressions lies in recognizing that they follow the same rules as ordinary fractions. Just as you need a common denominator to add or subtract numerical fractions, the same principle applies to rational expressions. This means finding a common multiple of the denominators involved. Simplification, another crucial aspect, involves canceling out common factors between the numerator and the denominator, similar to reducing a numerical fraction to its simplest form. Grasping these fundamental concepts is the first step in confidently tackling subtraction problems involving rational expressions.

Problem Breakdown: ${\frac{6-1}{2x^2} - \frac{3}{x}}$

Now, let's break down the specific problem at hand: ${\frac{6-1}{2x^2} - \frac{3}{x}}$. This expression involves the subtraction of two rational expressions. The first expression, ${\frac{6-1}{2x^2}}$, has a numerator that can be simplified immediately. The second expression, ${\frac{3}{x}}$, is already in its simplest form. To subtract these expressions, we need to find a common denominator. The denominators are ${2x^2}$ and ${x}$. Identifying the least common multiple (LCM) of these denominators is crucial for efficiently solving the problem. The LCM will be the simplest expression that both denominators can divide into evenly. Once we have the common denominator, we can rewrite each rational expression with this denominator, making the subtraction straightforward.

Finding the Common Denominator

The most critical step in subtracting rational expressions is identifying the common denominator. In our problem, the denominators are ${2x^2}$ and ${x}$. To find the least common denominator (LCD), we need to consider the factors of each denominator. The first denominator, ${2x^2}$, can be thought of as ${2 \\cdot x \\cdot x}$. The second denominator, ${x}$, is simply ${x}$. The LCD must include all the factors present in either denominator, raised to the highest power they appear in any of the denominators.

In this case, the LCD must include the constant factor 2, and the variable factor ${x}$ raised to the power of 2 (since ${x^2}$ is the highest power of ${x}$ present). Therefore, the LCD is ${2x^2}$. This means we need to rewrite both rational expressions so that they have this denominator. The first expression already has the denominator ${2x^2}$, so we don't need to change it. However, the second expression, ${\frac{3}{x}}$, needs to be adjusted. This adjustment will involve multiplying both the numerator and the denominator by a suitable factor to achieve the desired common denominator.

Rewriting Expressions with the Common Denominator

With the common denominator identified as ${2x^2}$, we now need to rewrite each rational expression to have this denominator. The first expression, ${\frac{6-1}{2x^2}}$, already has the correct denominator. Simplifying the numerator, we get ${\frac{5}{2x^2}}$. The second expression, ${\frac{3}{x}}$, needs to be rewritten. To change the denominator from ${x}$ to ${2x^2}$, we need to multiply it by ${2x}$. To keep the value of the expression the same, we must also multiply the numerator by the same factor. So, we multiply both the numerator and the denominator of ${\frac{3}{x}}$ by ${2x}$.

This gives us ${\frac{3 \\cdot 2x}{x \\cdot 2x} = \frac{6x}{2x^2}}$. Now, both rational expressions have the common denominator ${2x^2}$. We can proceed with the subtraction once both expressions are in this form. This step is crucial because it allows us to combine the numerators, which is the essence of subtracting fractions. Ensuring both rational expressions share the same denominator is a fundamental requirement for accurate subtraction.

Performing the Subtraction

Now that both rational expressions have the common denominator ${2x^2}$, we can perform the subtraction. We have ${\frac{5}{2x^2} - \frac{6x}{2x^2}}$. To subtract fractions with a common denominator, we simply subtract the numerators and keep the denominator the same. So, we subtract ${6x}$ from ${5}$, which gives us ${5 - 6x}$. The denominator remains ${2x^2}$.

Therefore, the result of the subtraction is ${\frac{5 - 6x}{2x^2}}$. This expression represents the difference between the two original rational expressions. It is important to check if this resulting expression can be further simplified. Simplification involves looking for common factors in the numerator and the denominator that can be canceled out. In this case, there are no common factors between ${5 - 6x}$ and ${2x^2}$, so the expression is already in its simplest form. Thus, ${\frac{5 - 6x}{2x^2}}$ is the final answer to the subtraction problem.

Simplifying the Result (If Possible)

After performing the subtraction, it's essential to simplify the resulting rational expression if possible. Simplification involves identifying and canceling out any common factors between the numerator and the denominator. This process is similar to reducing a numerical fraction to its lowest terms. In our case, the result of the subtraction was ${\frac{5 - 6x}{2x^2}}$. To simplify, we look for factors that are present in both the numerator and the denominator.

The numerator, ${5 - 6x}$, is a linear expression and doesn't have any obvious factors that would also be present in the denominator. The denominator, ${2x^2}$, has factors of 2 and ${x^2}$. Upon closer inspection, we can see that there are no common factors between the numerator and the denominator. This means that the expression ${\frac{5 - 6x}{2x^2}}$ is already in its simplest form. If we had found any common factors, we would divide both the numerator and the denominator by those factors to achieve the simplified expression. In this particular problem, however, no further simplification is possible.

Conclusion: Mastering Rational Expression Subtraction

In conclusion, subtracting rational expressions involves a series of steps that build upon fundamental algebraic principles. We began with the problem ${\frac{6-1}{2x^2} - \frac{3}{x}}$ and systematically worked through each stage. First, we simplified the initial expression by evaluating ${6 - 1}$ to get ${5}$ in the numerator of the first term. Next, we identified the common denominator, which was ${2x^2}$. We then rewrote each rational expression with this common denominator, adjusting the numerators accordingly. This involved multiplying the numerator and denominator of ${\frac{3}{x}}$ by ${2x}$ to obtain ${\frac{6x}{2x^2}}$.

After rewriting the expressions, we performed the subtraction by subtracting the numerators while keeping the common denominator. This resulted in the expression ${\frac{5 - 6x}{2x^2}}$. Finally, we checked for any possible simplifications but found that the expression was already in its simplest form. Therefore, the final answer to the problem is ${\frac{5 - 6x}{2x^2}}$. This process demonstrates that understanding the underlying principles of fraction manipulation and polynomial algebra is key to successfully subtracting rational expressions. By following these steps carefully, you can confidently tackle similar problems and deepen your understanding of algebraic operations.

In summary, mastering rational expression subtraction is not just about following steps; it's about understanding the 'why' behind each step. This understanding empowers you to adapt these techniques to various problems and build a stronger foundation in algebra. Remember, practice is key. The more you work with rational expressions, the more comfortable and proficient you will become. Keep exploring, keep questioning, and keep practicing!