Simplifying (2x)/(x^2-64) - 1/(x+8) A Step-by-Step Guide
Introduction
In the realm of mathematics, dealing with rational expressions is a fundamental skill. Rational expressions, which are essentially fractions involving polynomials, often appear in various branches of mathematics, including algebra, calculus, and beyond. In this detailed exploration, we'll delve into simplifying and solving a specific rational expression: (2x)/(x^2-64) - 1/(x+8). This problem provides an excellent opportunity to understand key concepts such as factoring, finding common denominators, simplifying expressions, and identifying restrictions on the variable. Our journey will not only involve solving the expression but also understanding the underlying principles that govern rational expressions.
This discussion aims to provide a comprehensive walkthrough, making the process clear and accessible for students and math enthusiasts alike. We will start by identifying the critical components of the expression, then move on to factoring the denominators, finding the least common denominator, and performing the necessary algebraic manipulations to simplify the expression. Finally, we will discuss any restrictions on the variable x that must be considered to ensure the validity of our solution. Understanding these steps is crucial for anyone looking to master algebraic manipulations and problem-solving in mathematics.
The ability to work with rational expressions is essential for more advanced topics in mathematics. Whether you're a student learning the basics of algebra or someone looking to brush up on your math skills, this guide will provide you with the necessary tools and understanding. We will break down each step, ensuring that you not only get the correct answer but also understand the 'why' behind each manipulation. By the end of this discussion, you should feel confident in your ability to tackle similar problems and appreciate the elegance and precision of algebraic techniques. Let's embark on this mathematical journey and demystify the process of simplifying and solving rational expressions.
Breaking Down the Expression: (2x)/(x^2-64)
To effectively address the given rational expression, the first step is to meticulously break down the expression: (2x)/(x^2-64) - 1/(x+8). Identifying its components and understanding their relationships is crucial for simplification. The expression consists of two main terms separated by a subtraction sign. The first term is a rational expression itself, (2x)/(x^2-64), where 2x is the numerator and x^2-64 is the denominator. The second term is 1/(x+8), another rational expression with a simpler structure. The key to simplifying this entire expression lies in our ability to manipulate these terms effectively.
Looking closely at the first term, (2x)/(x^2-64), we can immediately spot an opportunity for simplification. The denominator, x^2-64, is a difference of squares. Recognizing patterns like the difference of squares is a fundamental skill in algebra. The difference of squares pattern is expressed as a^2 - b^2 = (a - b)(a + b). In our case, x^2-64 fits this pattern perfectly, where a = x and b = 8. Thus, we can factor x^2-64 into (x - 8)(x + 8). This factorization is a critical step because it will allow us to find a common denominator with the second term, 1/(x+8). The ability to recognize and apply factoring techniques is essential for simplifying rational expressions.
Now, rewriting the first term with the factored denominator, we have (2x)/((x - 8)(x + 8)). This form reveals the structure of the expression more clearly and sets the stage for combining it with the second term. The second term, 1/(x+8), already has a relatively simple denominator. However, to subtract these two terms, we need a common denominator. The factored form of the first term's denominator, (x - 8)(x + 8), gives us a clear indication of what that common denominator should be. This detailed analysis of the components lays the groundwork for the subsequent steps, ensuring that we proceed with a clear understanding of the expression's structure. By carefully examining each part, we prepare ourselves for the algebraic manipulations required to simplify and solve the expression.
Factoring and Finding the Common Denominator
After breaking down the expression, factoring and finding the common denominator are the next crucial steps. We've already identified that the denominator of the first term, x^2-64, can be factored using the difference of squares pattern. As we discussed, x^2-64 factors into (x - 8)(x + 8). This factorization is pivotal because it reveals the underlying structure of the expression and helps us determine the least common denominator (LCD). Factoring not only simplifies individual terms but also allows us to combine them effectively.
With the denominator of the first term factored, we now have (2x)/((x - 8)(x + 8)) as the first term and 1/(x+8) as the second term. To subtract these two fractions, we need to find the LCD. The LCD is the smallest expression that is divisible by both denominators. In this case, the denominators are (x - 8)(x + 8) and (x + 8). The LCD will therefore include all unique factors from both denominators. Observing the two denominators, we see that the LCD is (x - 8)(x + 8). This is because it includes both factors present in the first denominator and the single factor present in the second denominator.
Now that we have the LCD, we need to rewrite each fraction with this common denominator. The first term, (2x)/((x - 8)(x + 8)), already has the LCD as its denominator, so we don't need to change it. However, the second term, 1/(x+8), needs to be adjusted. To get the LCD, (x - 8)(x + 8), in the denominator of the second term, we need to multiply both the numerator and the denominator of 1/(x+8) by (x - 8). This gives us (1 * (x - 8))/((x + 8) * (x - 8)), which simplifies to (x - 8)/((x - 8)(x + 8)). It's essential to multiply both the numerator and the denominator by the same expression to maintain the fraction's value. This process of finding the LCD and rewriting fractions is a cornerstone of working with rational expressions, enabling us to combine and simplify them effectively. The next step will involve performing the subtraction and simplifying the resulting expression.
Simplifying the Expression
With the common denominator in place, simplifying the expression becomes the next logical step. We have rewritten the original expression as (2x)/((x - 8)(x + 8)) - (x - 8)/((x - 8)(x + 8)). Now that both terms have the same denominator, we can subtract the numerators. This involves combining like terms and carefully managing the signs. When subtracting rational expressions, it's crucial to distribute the negative sign correctly to avoid errors.
Subtracting the numerators, we get (2x - (x - 8))/((x - 8)(x + 8)). Notice the parentheses around (x - 8) in the numerator; this is a critical detail. We need to distribute the negative sign across both terms inside the parentheses. Distributing the negative sign, we have 2x - x + 8 in the numerator. Combining the like terms, 2x and -x, simplifies to x. Thus, the numerator becomes x + 8. The expression now looks like (x + 8)/((x - 8)(x + 8)).
At this stage, we can observe a significant simplification opportunity. The numerator, (x + 8), is identical to one of the factors in the denominator. When the same factor appears in both the numerator and the denominator, we can cancel it out, provided that x ≠-8 (we'll discuss restrictions later). Cancelling the (x + 8) term from both the numerator and the denominator, we are left with 1/(x - 8). This is the simplified form of the original expression. Simplification not only makes the expression more manageable but also often reveals the underlying mathematical structure more clearly.
Through this process, we've seen how factoring, finding common denominators, subtracting numerators, and cancelling common factors work together to simplify a rational expression. The simplified expression, 1/(x - 8), is much easier to work with than the original. However, it's crucial to remember the restrictions on the variable x that arise from the original expression. These restrictions ensure that we don't divide by zero, which is undefined in mathematics. In the next section, we will delve into these restrictions and discuss their importance in the context of rational expressions. Understanding restrictions is just as important as simplifying the expression itself, as it ensures the mathematical validity of our result.
Identifying Restrictions on x
While simplifying rational expressions is a vital skill, identifying restrictions on x is equally crucial. Restrictions on x are values that would make the denominator of the original expression equal to zero, leading to an undefined result. In mathematics, division by zero is not permitted, so we must exclude these values from the domain of our expression. These restrictions stem from the original expression and must be considered even after simplification.
Looking back at the original expression, (2x)/(x^2-64) - 1/(x+8), we have two denominators to consider: x^2-64 and x+8. To find the restrictions, we set each denominator equal to zero and solve for x. First, let's consider x^2-64 = 0. This equation can be solved by factoring the left side as a difference of squares, which gives us (x - 8)(x + 8) = 0. Setting each factor equal to zero, we get x - 8 = 0 and x + 8 = 0. Solving these equations yields x = 8 and x = -8.
Next, let's consider the second denominator, x+8. Setting x+8 = 0, we solve for x and find x = -8. Notice that x = -8 appears as a restriction from both denominators, emphasizing its importance. The values x = 8 and x = -8 are the restrictions on x for this expression. This means that x cannot be 8 or -8; otherwise, we would be dividing by zero in the original expression. It's crucial to state these restrictions explicitly when simplifying rational expressions.
Even though we simplified the expression to 1/(x - 8), the restrictions from the original expression still apply. The simplified form does not show the restriction x = -8, but it is still a restriction because it makes the original denominator zero. This highlights a critical point: restrictions are determined by the original expression, not the simplified form. Therefore, the complete simplification includes both the simplified expression and the restrictions on x. In summary, the simplified expression is 1/(x - 8), with the restrictions x ≠8 and x ≠-8. Understanding and stating these restrictions ensures the mathematical correctness and completeness of our solution.
Conclusion
In conclusion, we have thoroughly explored the process of simplifying and solving the rational expression (2x)/(x^2-64) - 1/(x+8). Our journey began with a detailed breakdown of the expression, where we identified the key components and recognized opportunities for simplification. Factoring the denominator x^2-64 using the difference of squares pattern was a crucial step, leading us to rewrite the expression in a more manageable form. We then focused on finding the least common denominator, which allowed us to combine the two terms effectively.
The process of simplifying the expression involved subtracting the numerators and carefully handling the signs. By distributing the negative sign correctly and combining like terms, we arrived at the simplified numerator. We then recognized and cancelled out common factors between the numerator and the denominator, leading us to the simplified expression 1/(x - 8). This simplification demonstrates the power of algebraic manipulations in reducing complex expressions to their simplest forms. However, our task was not complete with just simplification.
Equally important was identifying and stating the restrictions on x. These restrictions, x ≠8 and x ≠-8, are values that would make the original denominators zero, resulting in an undefined expression. We emphasized that these restrictions must be considered from the original expression, even after simplification. By stating these restrictions, we ensure the mathematical validity and completeness of our solution. Understanding and applying these techniques is essential for anyone working with rational expressions in mathematics. The ability to simplify and solve such expressions is a foundational skill that underpins more advanced mathematical concepts. We hope this detailed exploration has provided a clear and comprehensive understanding of the process.
