Simplifying Rational Expressions: A Step-by-Step Guide

Have you ever stumbled upon a math problem that looks like a fraction filled with algebraic expressions and felt a little intimidated? Well, you're not alone! Rational expressions, which are essentially fractions with polynomials, can seem tricky at first. But don't worry, with a few key steps, you can simplify them like a pro. This guide will walk you through how to simplify a specific rational expression, providing you with the tools and understanding to tackle similar problems. Let's dive into the world of simplifying rational expressions and make math a little less daunting and a lot more fun!

Understanding Rational Expressions

Before we jump into the problem, let's make sure we're on the same page about what rational expressions are. At their core, rational expressions are fractions where the numerator and denominator are polynomials. Think of polynomials as expressions involving variables raised to non-negative integer powers, like $x^2$, $3x$, or even just a constant number like 5. When you combine these polynomials in a fraction, you get a rational expression. Now, why do we need to simplify them? Well, just like regular fractions, simplifying rational expressions makes them easier to work with and understand. A simplified expression is like a neatly organized room – everything is in its place, and it's much easier to find what you need. In mathematics, this means that simplified expressions are easier to use in further calculations and for understanding the relationship between variables. Simplifying these expressions often involves factoring polynomials, canceling out common factors, and applying the rules of fraction arithmetic. The goal is to express the rational expression in its most basic form, where the numerator and denominator have no common factors other than 1. This not only makes the expression cleaner but also helps in identifying any restrictions on the variable, such as values that would make the denominator zero.

The Problem: A Step-by-Step Approach

Let's tackle the specific problem at hand: simplify the expression $\frac{x^2+12 x+35}{x^2-49} \div \frac{x+7}{6 x-42}$. This looks like a mouthful, but we'll break it down step-by-step. Remember, the key to simplifying rational expressions lies in factoring and canceling common factors. Our first step is to deal with the division. Dividing by a fraction is the same as multiplying by its reciprocal. So, we rewrite the problem as $\frac{x^2+12 x+35}{x^2-49} \times \frac{6 x-42}{x+7}$. Now, we have a multiplication problem, which is easier to handle. Next, we need to factor each polynomial in the expression. This is where our algebra skills come into play. Factoring is like reverse multiplication – we're trying to find the expressions that, when multiplied together, give us the original polynomial. For instance, $x^2 + 12x + 35$ can be factored into $(x+5)(x+7)$. Similarly, $x^2 - 49$ is a difference of squares and factors into $(x-7)(x+7)$. And $6x - 42$ can be factored by taking out the common factor of 6, resulting in $6(x-7)$. Once we have factored each polynomial, we can rewrite the expression with all its factors. This sets the stage for the next crucial step: canceling common factors. Just like simplifying regular fractions, we can cancel out any factors that appear in both the numerator and the denominator. This is because any factor divided by itself is equal to 1, and multiplying by 1 doesn't change the value of the expression. By carefully factoring and canceling, we'll arrive at the simplest form of the rational expression.

Step 1: Rewrite Division as Multiplication

The first crucial step in simplifying the rational expression is to transform the division operation into multiplication. This is a fundamental rule in fraction arithmetic: dividing by a fraction is equivalent to multiplying by its reciprocal. So, instead of dividing by $\frac{x+7}{6 x-42}$, we multiply by its reciprocal, which is $\frac{6 x-42}{x+7}$. This transformation is essential because multiplication is often easier to work with when simplifying expressions. It allows us to combine terms and identify common factors more readily. By rewriting the expression, we set the stage for the next steps, which involve factoring and canceling. The original expression, $\frac{x^2+12 x+35}{x^2-49} \div \frac{x+7}{6 x-42}$, now becomes $\frac{x^2+12 x+35}{x^2-49} \times \frac{6 x-42}{x+7}$. This seemingly small change significantly simplifies the problem, making it more manageable. We've essentially flipped the second fraction and changed the division sign to a multiplication sign. This single step paves the way for us to break down the problem further and apply our factoring skills. It's like setting up the dominoes – once the first one falls (the division-to-multiplication conversion), the rest of the steps become clearer and easier to execute. This initial transformation is a cornerstone of simplifying rational expressions and a technique that you'll use repeatedly in similar problems.

Step 2: Factor Each Polynomial

The next vital step in simplifying our rational expression is to factor each polynomial. Factoring is like reverse engineering – we're trying to find the expressions that, when multiplied together, give us the original polynomial. This process is crucial because it allows us to identify common factors in the numerator and denominator, which we can then cancel out. Let's start with the numerator of the first fraction, $x^2 + 12x + 35$. We need to find two numbers that add up to 12 and multiply to 35. These numbers are 5 and 7, so we can factor the polynomial as $(x+5)(x+7)$. Now, let's move on to the denominator of the first fraction, $x^2 - 49$. This is a classic example of a difference of squares, which has a special factoring pattern: $a^2 - b^2 = (a-b)(a+b)$. In our case, $x^2 - 49$ can be factored as $(x-7)(x+7)$. Next, we consider the numerator of the second fraction, $6x - 42$. Here, we can factor out the greatest common factor, which is 6. This gives us $6(x-7)$. The denominator of the second fraction, $x+7$, is already in its simplest form and doesn't need further factoring. Now that we've factored each polynomial, we can rewrite the entire expression with its factored components. This step is like disassembling a machine into its individual parts – we can now see all the components clearly and how they fit together. The factored expression is $\frac{(x+5)(x+7)}{(x-7)(x+7)} \times \frac{6(x-7)}{x+7}$. With everything factored, we're in a prime position to identify and cancel common factors, bringing us closer to the simplified form of the expression.

Step 3: Cancel Common Factors

Now comes the exciting part: canceling common factors! This step is where all our hard work in factoring pays off. Just like simplifying regular fractions, we can cancel out any factors that appear in both the numerator and the denominator of our rational expression. This is because any factor divided by itself equals 1, and multiplying by 1 doesn't change the value of the expression. Looking at our factored expression, $\frac{(x+5)(x+7)}{(x-7)(x+7)} \times \frac{6(x-7)}{x+7}$, we can identify several common factors. We have $(x+7)$ in the numerator of the first fraction and the denominator of the first fraction, so we can cancel those out. We also have $(x-7)$ in the denominator of the first fraction and the numerator of the second fraction, so we can cancel those as well. After canceling these common factors, our expression simplifies to $\frac{(x+5)}{1} \times \frac{6}{x+7}$. Notice how much cleaner the expression looks now! We've eliminated the factors that were causing the most complexity. However, we're not quite done yet. We still need to multiply the remaining fractions together. Canceling common factors is like weeding a garden – we're removing the unnecessary elements to allow the essential parts to thrive. By carefully canceling, we've stripped away the complexity and revealed the underlying structure of the expression. This step not only simplifies the expression but also makes it easier to understand and work with in further calculations. It's a crucial technique in simplifying rational expressions and a skill that will serve you well in more advanced mathematical contexts.

Step 4: Multiply Remaining Fractions

With the common factors canceled, we're now ready to multiply the remaining fractions. This step is straightforward: we multiply the numerators together and the denominators together. After canceling, our expression looks like $\frac{(x+5)}{1} \times \frac{6}{x+7}$. To multiply these fractions, we multiply $(x+5)$ by 6 to get $6(x+5)$, and we multiply 1 by $(x+7)$ to get $(x+7)$. This gives us the new fraction $\frac{6(x+5)}{x+7}$. We can further simplify the numerator by distributing the 6, which means multiplying 6 by both terms inside the parentheses: $6 * x = 6x$ and $6 * 5 = 30$. So, the numerator becomes $6x + 30$. Our expression now looks like $\frac{6x + 30}{x+7}$. This is the simplified form of the original rational expression. Multiplying the remaining fractions is like putting the final pieces of a puzzle together. We've eliminated the complexity through factoring and canceling, and now we're combining the remaining elements to arrive at the final, simplified expression. This step solidifies our understanding of fraction arithmetic and how it applies to rational expressions. By carefully multiplying the numerators and denominators, we ensure that we're accurately representing the simplified form of the original expression. The result, $\frac{6x + 30}{x+7}$, is a much cleaner and easier-to-work-with version of the expression we started with.

Final Result

After meticulously working through each step, we've arrived at the simplified form of the rational expression. Let's recap our journey: we started with the expression $\frac{x^2+12 x+35}{x^2-49} \div \frac{x+7}{6 x-42}$, and through a series of transformations, we've simplified it to $\frac{6x + 30}{x+7}$. This final result is a testament to the power of factoring and canceling common factors. We began by rewriting the division as multiplication, which allowed us to work with a single expression. Then, we factored each polynomial, breaking them down into their fundamental components. This step was crucial for identifying the common factors that we could cancel. Canceling these factors simplified the expression significantly, making it much easier to manage. Finally, we multiplied the remaining fractions, combining the numerators and denominators to arrive at our simplified result. The simplified expression, $\frac{6x + 30}{x+7}$, is not only cleaner but also easier to understand and use in further mathematical operations. It represents the same value as the original expression but in a more concise and manageable form. This process highlights the importance of simplification in mathematics – it allows us to work with complex expressions more efficiently and accurately. By mastering these steps, you'll be well-equipped to tackle a wide range of rational expression problems. Remember, the key is to break down the problem into smaller, manageable steps, and to apply the fundamental principles of factoring and fraction arithmetic.

Conclusion

Simplifying rational expressions might seem challenging at first, but by following a step-by-step approach, you can master this important algebraic skill. We've seen how rewriting division as multiplication, factoring polynomials, canceling common factors, and multiplying remaining fractions can transform a complex expression into a much simpler form. Remember, practice is key! The more you work with rational expressions, the more comfortable you'll become with factoring and simplifying them. So, keep practicing, and you'll be simplifying rational expressions like a pro in no time! For further exploration and practice, you might find helpful resources on websites like Khan Academy's Algebra Section, which offers a wealth of lessons and exercises on rational expressions and other algebra topics.