Subtracting Rational Expressions Step By Step Solution

In the realm of algebraic manipulations, subtracting rational expressions is a fundamental skill. This article delves into the intricacies of subtracting rational expressions, providing a comprehensive guide to master this essential mathematical operation. We will illustrate the process through a step-by-step solution of the expression $\frac{1}{2 d^2+16 d}-\frac{5 d}{2 d^2-18 d}$, ensuring clarity and understanding.

Understanding Rational Expressions

To effectively subtract rational expressions, it's crucial to first grasp the concept of rational expressions themselves. A rational expression is essentially 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 multiplication, with non-negative integer exponents. Examples of rational expressions include $\frac{x+1}{x-2}$, $\frac{3x^2-2x+5}{x^2+4}$, and the expressions we'll be working with in this article, $\frac{1}{2 d^2+16 d}$ and $\frac{5 d}{2 d^2-18 d}$. The key to subtracting these expressions lies in finding a common denominator, a concept analogous to subtracting regular fractions. Just as we need a common denominator to subtract $\frac{1}{3}$ from $\frac{1}{2}$, we need a common denominator to subtract rational expressions. This common denominator allows us to combine the numerators while maintaining the integrity of the fraction. The process involves factoring the denominators, identifying the least common multiple (LCM), and then adjusting the numerators accordingly. This meticulous approach ensures that we perform the subtraction accurately and arrive at the simplified result. Mastering the art of subtracting rational expressions opens doors to solving more complex algebraic problems and is a crucial step in advancing one's mathematical proficiency. Furthermore, understanding rational expressions extends beyond the classroom, finding applications in various fields such as engineering, physics, and computer science, where modeling real-world phenomena often involves manipulating algebraic fractions. Therefore, dedicating time to thoroughly understand this concept is a worthwhile investment in one's mathematical journey.

Step 1: Factoring the Denominators

The initial step in subtracting rational expressions involves factoring the denominators. Factoring is the process of breaking down a polynomial into its constituent factors, which are expressions that multiply together to give the original polynomial. This step is crucial because it allows us to identify the least common denominator (LCD), which is essential for combining the fractions. Let's consider the denominators in our expression: $2d^2 + 16d$ and $2d^2 - 18d$. For the first denominator, $2d^2 + 16d$, we can factor out a common factor of $2d$. This gives us $2d(d + 8)$. This factorization reveals the building blocks of the denominator, highlighting the factors $2d$ and $(d + 8)$. Similarly, for the second denominator, $2d^2 - 18d$, we can also factor out a common factor of $2d$, resulting in $2d(d - 9)$. Again, we've successfully decomposed the denominator into its fundamental factors, $2d$ and $(d - 9)$. These factored forms are the key to finding the LCD. By identifying the individual factors within each denominator, we can construct the LCD by taking the highest power of each unique factor present. In this case, the unique factors are $2d$, $(d + 8)$, and $(d - 9)$. This process of factoring not only simplifies the denominators but also provides a clear path towards finding the LCD, which is the cornerstone of subtracting rational expressions. The ability to factor polynomials efficiently is a foundational skill in algebra, and mastering it is crucial for success in manipulating rational expressions and other algebraic concepts. Furthermore, factoring extends beyond simple polynomial manipulation; it is a fundamental tool in solving equations, simplifying expressions, and understanding the behavior of functions. Therefore, a solid understanding of factoring techniques is indispensable for anyone pursuing mathematics or related fields.

Step 2: Finding the Least Common Denominator (LCD)

After successfully factoring the denominators, the next crucial step is to identify the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. In simpler terms, it's the least common multiple (LCM) of the denominators. The LCD serves as the common ground upon which we can combine the fractions, enabling us to perform the subtraction. To find the LCD, we examine the factored forms of the denominators obtained in the previous step. Recall that we factored $2d^2 + 16d$ into $2d(d + 8)$ and $2d^2 - 18d$ into $2d(d - 9)$. Now, we identify all the unique factors present in these expressions. The unique factors are $2d$, $(d + 8)$, and $(d - 9)$. To construct the LCD, we take each unique factor and raise it to the highest power it appears in any of the denominators. In this case, each factor appears only once, so we simply multiply them together. Thus, the LCD is $2d(d + 8)(d - 9)$. This LCD now serves as the common denominator for both fractions. By having a common denominator, we can rewrite each fraction with the LCD as its denominator, which allows us to combine the numerators and perform the subtraction. Finding the LCD is a critical step in subtracting rational expressions, as it ensures that we are working with equivalent fractions that can be combined meaningfully. A clear understanding of how to determine the LCD is essential for mastering this operation and for simplifying complex algebraic expressions. Moreover, the concept of the LCD extends beyond rational expressions and is fundamental in various mathematical contexts, including working with fractions, solving equations, and simplifying expressions involving different denominators.

Step 3: Rewriting the Fractions with the LCD

With the LCD determined, the subsequent step involves rewriting each fraction with the LCD as its denominator. This process ensures that both fractions have a common base, allowing for the numerators to be combined through subtraction. To achieve this, we multiply both the numerator and the denominator of each fraction by the factors that are missing from its original denominator compared to the LCD. Let's revisit our original expression: $\frac{1}{2 d^2+16 d}-\frac{5 d}{2 d^2-18 d}$. We've already established that the LCD is $2d(d + 8)(d - 9)$. Now, let's focus on the first fraction, $\frac{1}{2 d^2+16 d}$. We know that the denominator $2d^2 + 16d$ factors into $2d(d + 8)$. Comparing this to the LCD, $2d(d + 8)(d - 9)$, we see that the factor $(d - 9)$ is missing. Therefore, we multiply both the numerator and the denominator of the first fraction by $(d - 9)$: $\frac{1}{2d(d+8)} \cdot \frac{(d-9)}{(d-9)} = \frac{d-9}{2d(d+8)(d-9)}$. Now, let's consider the second fraction, $\frac{5 d}{2 d^2-18 d}$. The denominator $2d^2 - 18d$ factors into $2d(d - 9)$. Comparing this to the LCD, we see that the factor $(d + 8)$ is missing. So, we multiply both the numerator and the denominator of the second fraction by $(d + 8)$: $\frac{5d}{2d(d-9)} \cdot \frac{(d+8)}{(d+8)} = \frac{5d(d+8)}{2d(d-9)(d+8)}$. By rewriting both fractions with the LCD, we've created equivalent fractions that share a common denominator. This crucial step sets the stage for the next phase, which involves subtracting the numerators while keeping the LCD intact. This method of rewriting fractions with a common denominator is a cornerstone of fraction arithmetic and extends beyond rational expressions to encompass any fraction manipulation. It is a fundamental skill in algebra and essential for simplifying complex expressions and solving equations involving fractions.

Step 4: Subtracting the Numerators

With both fractions now sharing the least common denominator (LCD), the next step involves subtracting the numerators. This is where we combine the numerators over the common denominator, effectively performing the subtraction operation. Recall that we've rewritten our expression as $\frac{d-9}{2d(d+8)(d-9)} - \frac{5d(d+8)}{2d(d-9)(d+8)}$. Since the denominators are now the same, we can subtract the numerators directly: $\frac{(d-9) - 5d(d+8)}{2d(d+8)(d-9)}$. Now, let's focus on simplifying the numerator. We have $(d - 9) - 5d(d + 8)$. First, we distribute the $-5d$ across the $(d + 8)$ term: $(d - 9) - (5d^2 + 40d)$. Next, we remove the parentheses, remembering to distribute the negative sign: $d - 9 - 5d^2 - 40d$. Finally, we combine like terms: $-5d^2 + d - 40d - 9$, which simplifies to $-5d^2 - 39d - 9$. So, our expression now becomes $\frac{-5d^2 - 39d - 9}{2d(d+8)(d-9)}$. This step of subtracting the numerators is a fundamental operation in fraction arithmetic. It combines the individual fractions into a single fraction, simplifying the expression and bringing us closer to the final result. The careful handling of signs and the correct application of the distributive property are crucial in this step to avoid errors. By meticulously simplifying the numerator, we ensure that our final answer is accurate and in its most reduced form. Moreover, the ability to subtract numerators efficiently is a key skill in algebra and is essential for solving equations, simplifying expressions, and performing various mathematical operations involving fractions and rational expressions. This step bridges the gap between individual fractions and a combined expression, showcasing the power of common denominators in simplifying complex algebraic problems.

Step 5: Simplifying the Result

The final step in subtracting rational expressions is to simplify the result. This involves checking if the resulting expression can be further simplified by factoring the numerator and/or the denominator and then canceling out any common factors. Let's revisit our expression: $\frac{-5d^2 - 39d - 9}{2d(d+8)(d-9)}$. First, we examine the numerator, $-5d^2 - 39d - 9$. We need to see if this quadratic expression can be factored. Factoring quadratic expressions can sometimes be challenging, but it's a crucial step in simplifying rational expressions. In this case, the numerator can be factored as $-(5d + 3)(d + 3)$. Now, let's look at the denominator, $2d(d + 8)(d - 9)$. We already have this in factored form. So, our expression now looks like this: $\frac{-(5d + 3)(d + 3)}{2d(d+8)(d-9)}$. Now, we check for any common factors between the numerator and the denominator. In this case, there are no common factors that can be canceled out. The factors in the numerator, $-(5d + 3)$ and $(d + 3)$, are different from the factors in the denominator, $2d$, $(d + 8)$, and $(d - 9)$. Therefore, the expression is already in its simplest form. While we couldn't simplify further in this particular example, it's important to always check for common factors after subtracting the numerators. Simplification is a key aspect of working with rational expressions, as it ensures that the final answer is presented in its most concise and understandable form. By factoring and canceling common factors, we reduce the complexity of the expression and make it easier to work with in future calculations. Moreover, simplifying rational expressions is a fundamental skill in algebra and is essential for solving equations, graphing functions, and performing various mathematical operations involving fractions and rational functions. The ability to simplify expressions efficiently is a hallmark of mathematical proficiency and is crucial for success in advanced mathematical studies.

Final Answer

Therefore, the simplified result of subtracting the rational expressions is: $\frac{-(5d + 3)(d + 3)}{2d(d+8)(d-9)}$ or $\frac{-5d^2 - 39d - 9}{2d(d+8)(d-9)}$.

In conclusion, subtracting rational expressions involves a series of steps: factoring the denominators, finding the least common denominator (LCD), rewriting the fractions with the LCD, subtracting the numerators, and simplifying the result. Mastering these steps is crucial for success in algebra and beyond. The example we worked through, $\frac{1}{2 d^2+16 d}-\frac{5 d}{2 d^2-18 d}$, illustrates this process in detail, providing a clear roadmap for tackling similar problems. Each step is essential, and a thorough understanding of each one contributes to a comprehensive grasp of the overall concept. Factoring allows us to identify the building blocks of the denominators, paving the way for finding the LCD. The LCD serves as the common ground upon which we can combine the fractions, ensuring that we are working with equivalent expressions. Rewriting the fractions with the LCD sets the stage for subtracting the numerators, which combines the individual fractions into a single expression. Finally, simplifying the result ensures that our answer is in its most concise and understandable form. This process not only simplifies the expression at hand but also reinforces fundamental algebraic skills that are applicable in various mathematical contexts. The ability to subtract rational expressions efficiently is a valuable asset in solving equations, simplifying complex expressions, and tackling advanced mathematical problems. Furthermore, the concepts and techniques involved in subtracting rational expressions extend beyond the classroom, finding applications in fields such as engineering, physics, and computer science, where modeling real-world phenomena often involves manipulating algebraic fractions. Therefore, investing time in mastering this skill is a worthwhile endeavor for anyone pursuing mathematics or related disciplines. By diligently practicing and applying these steps, you can confidently subtract rational expressions and unlock a deeper understanding of algebraic manipulations.