Simplifying The Rational Expression (8/(3x)) + ((x+4)/(x^2+3x-4)) Step-by-Step

Rational expressions are a cornerstone of algebraic manipulation, forming the basis for solving complex equations and understanding various mathematical concepts. A rational expression is essentially a fraction where the numerator and denominator are polynomials. Simplifying these expressions is a crucial skill in algebra, allowing for easier manipulation and solution-finding. This article provides a comprehensive guide on simplifying the sum of rational expressions, focusing on the specific example of combining $\frac{8}{3x}$ and $\frac{x+4}{x^2+3x-4}$ into a single, simplified expression. We will break down the process into manageable steps, ensuring a clear understanding of each stage. Mastering the simplification of rational expressions opens doors to advanced mathematical concepts and problem-solving techniques. From basic arithmetic operations to advanced calculus, the ability to manipulate and simplify these expressions is paramount. Before diving into the specifics of our example, let’s briefly review the fundamental principles that govern rational expressions. Remember, a rational expression is in its simplest form when the numerator and denominator have no common factors other than 1. Simplifying often involves factoring polynomials, identifying common factors, and canceling them out. This process not only makes the expression more manageable but also reveals its underlying structure and properties. Keep in mind that rational expressions, like fractions, have restrictions on their domain. The denominator cannot be equal to zero, as this would result in an undefined expression. Therefore, when simplifying, it’s essential to note any values of the variable that would make the denominator zero. These values are excluded from the domain of the expression. In the following sections, we will apply these principles to simplify the given expression, providing detailed explanations and examples along the way. This step-by-step approach will empower you to confidently tackle similar problems and deepen your understanding of rational expressions.

Step 1: Factoring the Denominators

The initial crucial step in simplifying the sum of rational expressions is factoring the denominators. This process is essential for identifying common factors and determining the least common denominator (LCD), which is needed to combine the fractions. In our example, we have the expression:

$\frac{8}{3x} + \frac{x+4}{x^2+3x-4}$

Let's focus on the denominators. The first denominator, $3x$, is already in its simplest form, as 3 is a prime number and $x$ is a variable. However, the second denominator, $x^2 + 3x - 4$, is a quadratic expression that can be factored. To factor this quadratic, we need to find two numbers that multiply to -4 and add up to 3. These numbers are 4 and -1. Therefore, we can rewrite the quadratic expression as:

$x^2 + 3x - 4 = (x + 4)(x - 1)$

Now, our expression looks like this:

$\frac{8}{3x} + \frac{x+4}{(x+4)(x-1)}$

Factoring the denominators has revealed a common factor of $(x + 4)$ in the second term. This is a significant observation that will help us simplify the expression further. Identifying common factors is a key skill in simplifying rational expressions, and it often requires a good understanding of factoring techniques. In this case, factoring the quadratic expression allowed us to see the common factor, which we can now use to simplify the expression. Moreover, factoring the denominators helps us to identify any restrictions on the variable $x$. In this example, we can see that $x$ cannot be 0 (from the first denominator), -4, or 1 (from the factored second denominator). These values would make the denominators zero, resulting in undefined expressions. Keeping track of these restrictions is crucial for a complete and accurate simplification. In the next step, we will use the factored denominators to determine the least common denominator (LCD) and combine the fractions. Understanding the importance of factoring is paramount, as it sets the stage for the subsequent steps in simplifying rational expressions. Remember to always look for opportunities to factor polynomials, as it often leads to significant simplifications and insights.

Step 2: Identifying the Least Common Denominator (LCD)

After successfully factoring the denominators, the next crucial step is identifying the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. It serves as the common ground for combining the fractions in our rational expression. In our example, we have the expression:

$\frac{8}{3x} + \frac{x+4}{(x+4)(x-1)}$

We've already factored the denominators, which are $3x$ and $(x+4)(x-1)$. To find the LCD, we need to consider all the unique factors present in the denominators and take the highest power of each factor. In this case, the unique factors are $3$, $x$, $(x+4)$, and $(x-1)$. The highest power of each factor is 1, as they each appear only once in the denominators. Therefore, the LCD is the product of these factors:

LCD = $3x(x+4)(x-1)$

Determining the LCD is a critical step because it allows us to rewrite each fraction with a common denominator, making it possible to add or subtract them. The LCD ensures that we are working with equivalent fractions, maintaining the integrity of the expression. Understanding the concept of the LCD is not just about memorizing a procedure; it's about grasping the fundamental principle of finding a common multiple. Just as we find a common denominator when adding numerical fractions, we find the LCD when adding rational expressions. This common denominator allows us to combine the numerators while maintaining the correct proportions. Furthermore, the LCD is essential for identifying any restrictions on the variable. As we noted earlier, the denominators cannot be equal to zero. Therefore, we can see from the LCD that $x$ cannot be 0, -4, or 1. These values would make the LCD zero, resulting in undefined expressions. It's crucial to keep these restrictions in mind as we simplify the expression. In the next step, we will rewrite each fraction with the LCD as its denominator. This involves multiplying the numerator and denominator of each fraction by the factors needed to obtain the LCD. By mastering the concept of the LCD, you'll be well-equipped to tackle more complex rational expressions and algebraic manipulations. The LCD is a fundamental tool in the arsenal of any algebra student, and a solid understanding of it will pave the way for success in more advanced topics.

Step 3: Rewriting Fractions with the LCD

With the least common denominator (LCD) identified, the next step is to rewrite each fraction in the expression with the LCD as its new denominator. This process involves multiplying both the numerator and the denominator of each fraction by the factors needed to obtain the LCD. This ensures that we are creating equivalent fractions, which is essential for maintaining the value of the expression while allowing us to combine them. Recall our expression:

$\frac{8}{3x} + \frac{x+4}{(x+4)(x-1)}$

And our LCD:

LCD = $3x(x+4)(x-1)$

Let's rewrite the first fraction, $\frac{8}{3x}$. To get the LCD, we need to multiply the denominator $3x$ by $(x+4)(x-1)$. Therefore, we must also multiply the numerator by the same factors:

$\frac{8}{3x} \cdot \frac{(x+4)(x-1)}{(x+4)(x-1)} = \frac{8(x+4)(x-1)}{3x(x+4)(x-1)}$

Now, let's rewrite the second fraction, $\frac{x+4}{(x+4)(x-1)}$. To get the LCD, we need to multiply the denominator $(x+4)(x-1)$ by $3x$. Therefore, we must also multiply the numerator by $3x$:

$\frac{x+4}{(x+4)(x-1)} \cdot \frac{3x}{3x} = \frac{3x(x+4)}{3x(x+4)(x-1)}$

Now, our expression looks like this:

$\frac{8(x+4)(x-1)}{3x(x+4)(x-1)} + \frac{3x(x+4)}{3x(x+4)(x-1)}$

Rewriting fractions with the LCD is a fundamental technique in adding and subtracting rational expressions. It allows us to combine the numerators over a common denominator, which is the key to simplifying the expression. This process is analogous to finding a common denominator when adding numerical fractions. By multiplying the numerator and denominator by the appropriate factors, we ensure that the value of the fraction remains unchanged while allowing us to combine them. It's crucial to be meticulous in this step, ensuring that you multiply both the numerator and denominator by the correct factors. A mistake in this step can lead to an incorrect simplification of the expression. Furthermore, rewriting fractions with the LCD reinforces the importance of understanding equivalent fractions. We are essentially creating fractions that look different but have the same value, which is a core concept in algebra. In the next step, we will combine the numerators over the common denominator and simplify the resulting expression. Mastering this step of rewriting fractions with the LCD is essential for simplifying rational expressions and solving algebraic equations. It's a skill that will be used repeatedly in more advanced mathematical topics, making it a valuable tool in your mathematical arsenal.

Step 4: Combining the Numerators

Having rewritten the fractions with the least common denominator (LCD), the next step is to combine the numerators. This involves adding or subtracting the numerators while keeping the common denominator. It's a crucial step in simplifying the expression, as it brings us closer to a single, simplified fraction. Recall our expression after rewriting with the LCD:

$\frac{8(x+4)(x-1)}{3x(x+4)(x-1)} + \frac{3x(x+4)}{3x(x+4)(x-1)}$

Now we can combine the numerators over the common denominator:

$\frac{8(x+4)(x-1) + 3x(x+4)}{3x(x+4)(x-1)}$

This step essentially combines the two fractions into one, making it easier to simplify the expression further. However, the numerator is still in a complex form, and we need to expand and simplify it. Let's focus on the numerator:

$8(x+4)(x-1) + 3x(x+4)$

First, we expand the product $(x+4)(x-1)$:

$(x+4)(x-1) = x^2 - x + 4x - 4 = x^2 + 3x - 4$

Now, substitute this back into the numerator:

$8(x^2 + 3x - 4) + 3x(x+4)$

Next, distribute the 8 and the $3x$:

$8x^2 + 24x - 32 + 3x^2 + 12x$

Finally, combine like terms:

$8x^2 + 3x^2 + 24x + 12x - 32 = 11x^2 + 36x - 32$

Now, our expression looks like this:

$\frac{11x^2 + 36x - 32}{3x(x+4)(x-1)}$

Combining the numerators is a fundamental step in adding or subtracting fractions, whether they are numerical or rational. It's the process of bringing the fractions together under a common denominator, allowing us to work with a single fraction. This step often requires careful expansion and simplification of the numerator, as we saw in our example. It's essential to pay attention to the order of operations and combine like terms correctly. Furthermore, combining the numerators sets the stage for further simplification, such as factoring the numerator and canceling common factors with the denominator. This is the next step in our process, and it will allow us to arrive at the simplest form of the rational expression. In the next step, we will explore factoring the numerator and canceling common factors to achieve the final simplified expression. Mastering the process of combining numerators is a crucial skill in algebra and will serve you well in more advanced mathematical studies.

Step 5: Simplifying the Expression

After combining the numerators, the final step is to simplify the expression as much as possible. This typically involves factoring the numerator and denominator and then canceling out any common factors. This process reduces the expression to its simplest form, making it easier to work with and understand. Recall our expression after combining the numerators:

$\frac{11x^2 + 36x - 32}{3x(x+4)(x-1)}$

Now, let's focus on factoring the numerator, $11x^2 + 36x - 32$. This is a quadratic expression, and we need to find two binomials that multiply to give this quadratic. Factoring quadratics can sometimes be challenging, but with practice, it becomes more manageable. We are looking for two numbers that multiply to $11 \cdot -32 = -352$ and add up to 36. These numbers are 44 and -8. We can rewrite the middle term using these numbers:

$11x^2 + 36x - 32 = 11x^2 + 44x - 8x - 32$

Now, we can factor by grouping:

$11x^2 + 44x - 8x - 32 = 11x(x + 4) - 8(x + 4)$

Factor out the common factor $(x + 4)$:

$11x(x + 4) - 8(x + 4) = (11x - 8)(x + 4)$

Now, our expression looks like this:

$\frac{(11x - 8)(x + 4)}{3x(x+4)(x-1)}$

We can see a common factor of $(x + 4)$ in both the numerator and the denominator. We can cancel this common factor out:

$\frac{(11x - 8)(x + 4)}{3x(x+4)(x-1)} = \frac{11x - 8}{3x(x-1)}$

Now, the expression is in its simplest form. There are no more common factors to cancel out. Therefore, the simplified expression is:

$\frac{11x - 8}{3x(x-1)}$

Simplifying the expression by factoring and canceling common factors is a crucial step in working with rational expressions. It allows us to reduce the expression to its most manageable form, making it easier to use in further calculations or analysis. Factoring the numerator and denominator is a skill that requires practice and a good understanding of factoring techniques. It's essential to look for common factors and cancel them out to achieve the simplest form. Furthermore, simplifying the expression often reveals the underlying structure and properties of the expression. It can also help in identifying any restrictions on the variable, as we discussed earlier. In our example, we saw that $x$ cannot be 0, -4, or 1. These restrictions are important to keep in mind when working with the simplified expression. By mastering the process of simplifying rational expressions, you'll be well-equipped to tackle more complex algebraic problems and gain a deeper understanding of mathematical concepts. This skill is essential for success in algebra and beyond, making it a valuable tool in your mathematical journey.

In this article, we have explored a detailed, step-by-step process for simplifying rational expressions, using the example of combining $\frac{8}{3x}$ and $\frac{x+4}{x^2+3x-4}$. We began by factoring the denominators, a crucial step for identifying common factors and determining the LCD. Next, we identified the least common denominator (LCD), which allowed us to rewrite each fraction with a common denominator. We then rewrote the fractions with the LCD, ensuring that we were working with equivalent fractions. This was followed by combining the numerators, a step that involved expanding and simplifying the resulting expression. Finally, we simplified the expression by factoring the numerator and canceling out any common factors with the denominator.

This process, while detailed, is a fundamental skill in algebra. Mastering the simplification of rational expressions opens doors to more advanced mathematical concepts and problem-solving techniques. From basic arithmetic operations to advanced calculus, the ability to manipulate and simplify these expressions is paramount. Rational expressions are more than just algebraic fractions; they are building blocks for understanding functions, equations, and various mathematical models. The skills you develop in simplifying them will serve you well in many areas of mathematics and related fields.

Remember, the key to success in simplifying rational expressions lies in practice and a solid understanding of the underlying principles. Take the time to work through examples, and don't be afraid to make mistakes – they are valuable learning opportunities. With perseverance and a clear understanding of the steps involved, you can confidently tackle any rational expression and unlock its hidden simplicity. As you continue your mathematical journey, the ability to simplify rational expressions will become an invaluable asset, empowering you to explore more complex and fascinating mathematical landscapes. The journey of simplifying rational expressions is not just about finding the answer; it's about developing a deeper understanding of algebraic manipulation and problem-solving strategies. These skills will not only help you in mathematics but also in various other disciplines that require logical reasoning and analytical thinking.