Add Rational Expressions: A Step-by-Step Guide
Hey everyone! Today, we're diving into the world of rational expressions and tackling the task of adding them together. Specifically, we'll be walking through a problem where we need to combine two fractions with polynomial expressions in both the numerator and denominator. It might seem a bit daunting at first, but trust me, by breaking it down into manageable steps, you'll be a pro in no time! So, grab your pencils, and let's get started!
Understanding Rational Expressions
Before we jump into the problem, let's quickly recap what rational expressions are. Think of them as fractions where the numerator and denominator are polynomials. Polynomials, as you might remember, are expressions involving variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Examples include 3x^2 + 2x - 1, 5x^4 - 7, and even just a simple x or a constant number like 7. When we have one polynomial divided by another, we have a rational expression. Adding rational expressions is a fundamental operation in algebra, often encountered in simplifying complex equations and solving various mathematical problems. The core concept revolves around finding a common denominator, much like adding regular fractions. This common denominator allows us to combine the numerators and simplify the result. The process involves factoring the denominators, identifying the least common multiple (LCM), and adjusting the numerators accordingly. Once the expressions have a common denominator, adding them becomes straightforward, and the final step often involves simplifying the resulting expression.
The Problem at Hand
Okay, let's get to the specific problem we're going to solve. We have the following expression:
rac{2x}{3x - 4} + rac{x - 1}{2x - 2}
Our goal is to add these two rational expressions together and express the result in a simplified form. We also need to figure out what should go in the [?] and the boxes in the final expression:
rac{7x^2 - 11x + [?]}{(2x + oxed{ })(3x + oxed{ })}
This problem is a classic example of adding rational expressions, requiring us to find a common denominator, adjust the numerators, combine like terms, and potentially factor the result. It's a great exercise for honing our algebraic skills and understanding the nuances of polynomial manipulation. We will methodically work through each step, explaining the reasoning and techniques involved, ensuring a clear understanding of the process. So, let's put on our math hats and start unraveling this expression!
Step 1: Factoring the Denominators
The first crucial step in adding rational expressions is to factor the denominators. Factoring helps us identify common factors and determine the least common denominator (LCD). Looking at our expression, the first denominator, 3x - 4, is already in its simplest form – it can't be factored further. However, the second denominator, 2x - 2, can be factored. Notice that both terms have a common factor of 2. So, we can factor out the 2:
2x - 2 = 2(x - 1)
Now, our expression looks like this:
rac{2x}{3x - 4} + rac{x - 1}{2(x - 1)}
Factoring is a fundamental skill in algebra, and mastering it is key to simplifying expressions and solving equations. It allows us to rewrite expressions in a more manageable form, revealing hidden structures and relationships. In this case, factoring the denominator 2x - 2 not only simplifies the expression but also helps us in the next step of finding the least common denominator. By identifying common factors and prime factors, we can construct the LCD efficiently. Factoring plays a pivotal role in various mathematical operations, including solving quadratic equations, simplifying rational expressions, and working with polynomials. So, let's make sure we have a solid grasp of factoring techniques as we move forward in our mathematical journey.
Step 2: Finding the Least Common Denominator (LCD)
Now that we've factored the denominators, we need to find the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. In our case, the denominators are 3x - 4 and 2(x - 1). To find the LCD, we need to consider all the unique factors present in the denominators and take the highest power of each factor.
- The first denominator has a factor of
(3x - 4). Since it appears only once, we include it in the LCD. - The second denominator has a factor of
2. We also include this in the LCD. - The second denominator also has a factor of
(x - 1). We include this as well.
Therefore, the LCD is the product of these factors:
LCD = 2(3x - 4)(x - 1)
Finding the least common denominator is a crucial step in adding or subtracting fractions, whether they are simple numerical fractions or rational expressions. The LCD ensures that we can combine the fractions by having a common base. It's like finding a common unit of measurement before adding lengths – we need the denominators to be the same before we can add the numerators. The process of identifying the LCD involves careful examination of the denominators, breaking them down into their prime factors, and then constructing the LCD by taking the highest power of each factor. This skill is essential for simplifying complex expressions and solving equations involving fractions. So, with our LCD in hand, we're ready to move on to the next step of adjusting the numerators.
Step 3: Adjusting the Numerators
With the LCD determined, the next step is to rewrite each fraction with the LCD as its denominator. This involves multiplying both the numerator and denominator of each fraction by the factors needed to obtain the LCD. Let's start with the first fraction, 2x / (3x - 4). To get the LCD, 2(3x - 4)(x - 1), in the denominator, we need to multiply both the numerator and denominator by 2(x - 1):
rac{2x}{3x - 4} * rac{2(x - 1)}{2(x - 1)} = rac{4x(x - 1)}{2(3x - 4)(x - 1)}
Now, let's move on to the second fraction, (x - 1) / 2(x - 1). To get the LCD, we need to multiply both the numerator and denominator by (3x - 4):
rac{x - 1}{2(x - 1)} * rac{(3x - 4)}{(3x - 4)} = rac{(x - 1)(3x - 4)}{2(3x - 4)(x - 1)}
Now, both fractions have the same denominator, which is the LCD. Adjusting the numerators is a critical step in the process of adding rational expressions. It ensures that we are adding equivalent fractions, maintaining the value of the original expression. This step often involves careful multiplication and distribution to correctly transform the numerators. The key is to identify what factors are missing in the original denominator compared to the LCD and then multiply both the numerator and denominator by those missing factors. This process sets the stage for combining the numerators in the next step, leading us closer to the final simplified expression. So, with our adjusted numerators, we're ready to add the fractions together.
Step 4: Adding the Numerators
Now that both fractions have the same denominator, we can add the numerators. We simply add the numerators together and keep the common denominator:
rac{4x(x - 1)}{2(3x - 4)(x - 1)} + rac{(x - 1)(3x - 4)}{2(3x - 4)(x - 1)} = rac{4x(x - 1) + (x - 1)(3x - 4)}{2(3x - 4)(x - 1)}
Next, we need to expand and simplify the numerator. Let's start by expanding the products:
4x(x - 1) = 4x^2 - 4x
(x - 1)(3x - 4) = 3x^2 - 4x - 3x + 4 = 3x^2 - 7x + 4
Now, substitute these back into the numerator:
4x^2 - 4x + 3x^2 - 7x + 4
Combine like terms:
(4x^2 + 3x^2) + (-4x - 7x) + 4 = 7x^2 - 11x + 4
So, our expression now looks like this:
rac{7x^2 - 11x + 4}{2(3x - 4)(x - 1)}
Adding the numerators is a pivotal step in combining rational expressions. It's where we bring together the adjusted numerators over the common denominator. This process often involves expanding products, distributing terms, and then combining like terms to simplify the resulting expression. Careful attention to detail is crucial in this step to avoid errors in the algebraic manipulation. Once the numerators are added and simplified, we have a single fraction representing the sum of the original rational expressions. This simplified numerator sets the stage for the final step of factoring and simplifying the entire expression, if possible. So, with our combined numerator, we're on the verge of reaching the final answer.
Step 5: Factoring and Simplifying (if possible)
Now, let's see if we can factor the numerator, 7x^2 - 11x + 4. We're looking for two binomials that multiply to give us this quadratic expression. After some trial and error, or using factoring techniques, we find that:
7x^2 - 11x + 4 = (7x - 4)(x - 1)
So, our expression becomes:
rac{(7x - 4)(x - 1)}{2(3x - 4)(x - 1)}
Now, we can see that there's a common factor of (x - 1) in both the numerator and the denominator. We can cancel this common factor:
rac{(7x - 4)(x - 1)}{2(3x - 4)(x - 1)} = rac{7x - 4}{2(3x - 4)}
We can leave the denominator as it is or distribute the 2:
rac{7x - 4}{6x - 8}
Factoring and simplifying is the final polish in the process of adding rational expressions. It allows us to express the result in its simplest form, revealing any hidden cancellations or relationships. Factoring the numerator and denominator can uncover common factors that can be eliminated, reducing the expression to its most basic form. This step often requires a good understanding of factoring techniques and the ability to recognize patterns. Simplifying the expression not only makes it more concise but also makes it easier to work with in further calculations or applications. So, with our expression fully simplified, we've successfully navigated the process of adding rational expressions.
Step 6: Matching the Form and Finding the Missing Values
Finally, let's go back to the original question and match our result to the given form:
rac{7x^2 - 11x + [?]}{(2x + oxed{ })(3x + oxed{ })}
We found that the numerator is 7x^2 - 11x + 4, so the missing value [?] is 4. Now, let's look at the denominator. We factored the original denominators as 3x - 4 and 2(x - 1). The given form has (2x + oxed{ }) and (3x + oxed{ }). Comparing these, we can see that:
3x - 4matches(3x + oxed{ }), so the missing value in the box is-4.2(x - 1) = 2x - 2. We need to rewrite this as2x + oxed{ }, so the missing value in the box is-2.
Therefore, the final answer is:
rac{7x^2 - 11x + 4}{(2x - 2)(3x - 4)}
Matching the form and finding the missing values is the final step in ensuring we've fully answered the question. It requires us to connect our simplified result back to the original problem statement and identify any missing components. This step often involves comparing expressions, matching coefficients, and solving for unknown values. It's a crucial step in verifying that we've addressed all aspects of the problem and provided a complete solution. This final check ensures that our answer is not only mathematically correct but also presented in the requested format. So, with our missing values identified, we can confidently say that we've conquered this rational expression problem!
Conclusion
Adding rational expressions might seem tricky at first, but by breaking it down into these steps – factoring, finding the LCD, adjusting numerators, adding, and simplifying – it becomes much more manageable. Remember, practice makes perfect! The more you work with these types of problems, the more comfortable you'll become with the process. You've got this! Understanding the step-by-step approach to adding rational expressions is crucial for mastering algebraic manipulations. By factoring denominators, finding the least common denominator, adjusting numerators, combining like terms, and simplifying the result, we can effectively tackle complex expressions. This process not only enhances our algebraic skills but also provides a solid foundation for advanced mathematical concepts. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries!