Simplifying Rational Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the exciting world of rational expressions, specifically focusing on how to divide them. Don't worry, it's not as scary as it sounds! In fact, it's pretty straightforward once you get the hang of it. We'll break down the problem step-by-step, making sure you grasp every concept. We will use the following expression: $\frac{2 x^2-11 x-21}{5 x^2+16 x+3} \div \frac{5 x-35}{5 x+1}$. Ready to conquer this mathematical challenge? Let's go! This article will guide you through the process of simplifying this rational expression, providing a detailed explanation of each step involved. This is a common problem in algebra, so understanding it will boost your skills. We'll be using factoring, simplifying, and remembering how to divide fractions. So grab your pencils, and let's get started. Understanding these concepts helps you solve more complex math problems. Understanding the order of operations and properties of exponents is also essential. Remember that practice is key, and the more you practice these types of problems, the more confident you'll become.

Step 1: Understand the Problem

First, we need to understand what we're dealing with. We're given a division problem involving two rational expressions. Remember that a rational expression is simply a fraction where the numerator and denominator are polynomials. Our goal is to simplify this expression as much as possible. This means we'll perform the division and then cancel out any common factors in the numerator and denominator. We will use the following expression: $\frac{2 x^2-11 x-21}{5 x^2+16 x+3} \div \frac{5 x-35}{5 x+1}$. Before we start, let's refresh our memory on how to divide fractions. Dividing fractions is the same as multiplying by the reciprocal of the second fraction. So, instead of dividing, we'll multiply the first expression by the reciprocal of the second. This transforms the division problem into a multiplication problem, which is usually easier to handle. Always double-check your work at each step to avoid errors. The core concept here is to rewrite the division as multiplication by the reciprocal. This simple trick unlocks the path to solving the problem. Keep in mind that understanding each step is vital to solving more complex problems down the line. We are transforming the division into multiplication by the reciprocal. This is a crucial first step.

Step 2: Rewrite the Division as Multiplication

As mentioned earlier, dividing by a fraction is the same as multiplying by its reciprocal. So, our first step is to rewrite the expression. We'll keep the first rational expression the same and multiply it by the reciprocal of the second rational expression. Let's write that out. The reciprocal of $\frac{5 x-35}{5 x+1}$ is $\frac{5 x+1}{5 x-35}$. Therefore, our problem becomes: $\frac{2 x^2-11 x-21}{5 x^2+16 x+3} \times \frac{5 x+1}{5 x-35}$. See? Now we have a multiplication problem! This is a much more manageable format. Make sure you correctly identify the reciprocal. A common mistake is flipping the wrong fraction. Remember, we are only changing the operation and inverting the second fraction. This step sets us up for factoring and simplifying. Now, it's a matter of simplifying. We transformed the division problem into a multiplication problem, making it easier to solve. The expression is now ready for factoring.

Step 3: Factor the Expressions

Next up, we need to factor all the polynomials in the numerators and denominators. Factoring is the process of breaking down a polynomial into a product of simpler expressions. This will allow us to identify and cancel out any common factors. Factoring might seem tricky at first, but with practice, you'll become a pro. We need to factor the following: $2x^2 - 11x - 21$, $5x^2 + 16x + 3$, $5x + 1$, and $5x - 35$. Let's start with $2x^2 - 11x - 21$. This quadratic can be factored into $(2x + 3)(x - 7)$. Next, $5x^2 + 16x + 3$ factors into $(5x + 1)(x + 3)$. The expression $5x + 1$ is already in its simplest form. Finally, $5x - 35$ can be factored into $5(x - 7)$. So, our expression now looks like this: $\frac{(2x + 3)(x - 7)}{(5x + 1)(x + 3)} \times \frac{5x + 1}{5(x - 7)}$. Remember to look for the greatest common factor (GCF) when factoring. Factoring each expression is a critical step, so make sure you do it correctly. This step is about breaking down each polynomial into simpler components, making it easier to simplify the overall expression. We've taken each polynomial and broken it down into its factors. This is a vital step in solving the problem.

Step 4: Cancel Common Factors

Now comes the fun part: canceling out common factors! Once we've factored all the polynomials, we can look for factors that appear in both the numerator and the denominator. When we find them, we can cancel them out. In our expression, $\frac{(2x + 3)(x - 7)}{(5x + 1)(x + 3)} \times \frac{5x + 1}{5(x - 7)}$, we can see that $(x - 7)$ appears in the numerator and the denominator, so we can cancel it. Also, $(5x + 1)$ appears in both the numerator and denominator, so we can cancel that as well. After canceling the common factors, our expression becomes: $\frac{2x + 3}{x + 3} \times \frac{1}{5}$. Canceling is a crucial step for simplifying rational expressions. Always double-check to make sure you've canceled all common factors. This step simplifies the expression by eliminating common factors. The key here is to identify and cancel any common factors. This simplifies the expression, making the solution clearer.

Step 5: Simplify the Expression

Almost there, guys! After canceling the common factors, we're left with $\frac{2x + 3}{x + 3} \times \frac{1}{5}$. Now, let's multiply the remaining terms. Multiplying the numerators, we get $1 * (2x + 3) = 2x + 3$. Multiplying the denominators, we get $5 * (x + 3) = 5x + 15$. So, our simplified expression is $\frac{2x + 3}{5x + 15}$. This is the simplest form of the original expression. Always simplify your answer as much as possible. After canceling the common factors, we're left with $\frac{2x + 3}{5x + 15}$. This is the simplified expression. Always double-check your final answer to make sure you didn't miss any steps. This final simplification gives us our final answer. We have simplified the expression step by step. This is the final answer.

Step 6: The Final Answer

So, after all that hard work, the simplified form of $\frac{2 x^2-11 x-21}{5 x^2+16 x+3} \div \frac{5 x-35}{5 x+1}$ is $\frac{2x + 3}{5x + 15}$. Congratulations! You've successfully simplified a complex rational expression. Remember, this is a combination of several skills. Always take your time and double-check your work. Practice more problems to become more confident. Also, understanding the concepts will help you solve more complex math problems. Keep up the great work, and you'll become a pro in no time! Always remember the steps we followed today: rewrite the division, factor, cancel, and simplify. Math can be tricky, but it's rewarding when you get it right. Always be patient with yourself and celebrate your achievements! Now you're equipped to solve similar problems. Always go back and check your work to avoid making simple mistakes.

Additional Tips

• Practice Regularly: The more you practice, the better you'll become. Work through different types of problems to solidify your understanding.
• Master Factoring: Factoring is a key skill for simplifying rational expressions. Make sure you're comfortable with different factoring techniques.
• Understand the Rules: Remember the rules for dividing fractions and canceling common factors.
• Check Your Work: Always double-check your work to avoid errors.
• Ask for Help: Don't hesitate to ask your teacher or classmates for help if you're stuck.

By following these steps and tips, you'll be well on your way to mastering the simplification of rational expressions! Keep practicing, and you'll find that these problems become easier with time.