Dividing Rational Expressions: A Step-by-Step Solution

In this article, we'll walk through the process of dividing rational expressions and simplifying the result to its reduced form. We'll tackle the specific problem: What is the quotient in reduced form of the rational expressions $\frac{x^2-4}{x+3} \div \frac{x^2-4 x+4}{4 x+12}$? This is a common type of problem in algebra, and understanding the steps involved is crucial for mastering rational expressions. Let's dive in!

Understanding Rational Expressions and Division

Before we jump into solving the problem, let's quickly review what rational expressions are and how division works in this context. Rational expressions are essentially fractions where the numerator and denominator are polynomials. Think of them as algebraic fractions. Dividing rational expressions is similar to dividing regular fractions, but with the added complexity of dealing with polynomials. The key concept to remember is that dividing by a fraction is the same as multiplying by its reciprocal. This is the golden rule we'll use to solve our problem.

When we talk about dividing rational expressions, we are essentially looking at simplifying complex fractions. These expressions often involve variables and require us to factorize polynomials to find common factors that can be canceled out. The goal is to express the final answer in its simplest form, also known as the reduced form. This involves several steps: first, we need to flip the second fraction and change the division to multiplication. Then, we factorize all the polynomials in both the numerator and the denominator. After factorization, we look for common factors between the numerator and the denominator and cancel them out. This process of simplifying rational expressions is a cornerstone of algebraic manipulation and is used extensively in higher mathematics, including calculus and differential equations. Understanding these steps thoroughly will not only help in solving such problems but will also build a solid foundation for more advanced mathematical concepts. Therefore, paying close attention to each step and practicing with different examples is highly recommended to gain proficiency in this area. Let's now apply these concepts to our specific problem and see how we can find the quotient in reduced form.

Step 1: Rewrite the Division as Multiplication

The first step in solving this problem is to rewrite the division as multiplication by the reciprocal. This means we flip the second fraction and change the division sign to a multiplication sign. So, we have:

$$\frac{x^2-4}{x+3} \div \frac{x^2-4 x+4}{4 x+12} = \frac{x^2-4}{x+3} \times \frac{4 x+12}{x^2-4 x+4}$$

This transformation is crucial because it allows us to combine the fractions into a single expression, making it easier to simplify. By changing the division problem into a multiplication problem, we set the stage for factoring and canceling out common terms. This step simplifies the process significantly and allows us to apply familiar techniques of fraction manipulation. It’s important to remember that this step is based on the fundamental principle of fraction division, where dividing by a fraction is equivalent to multiplying by its inverse. Therefore, understanding this basic principle is key to successfully navigating through the rest of the problem. Once we have rewritten the expression, we can move forward to the next step, which involves factoring the polynomials. This will help us identify common factors that can be canceled out, leading us closer to the final simplified answer.

Step 2: Factor the Polynomials

Now, we need to factor each polynomial in the expression. Factoring is the process of breaking down a polynomial into its constituent factors. This will help us identify common factors that can be canceled out later.

• $x^2 - 4$: This is a difference of squares, which can be factored as $(x + 2)(x - 2)$.
• $x + 3$: This is already in its simplest form and cannot be factored further.
• $4x + 12$: We can factor out a 4, resulting in $4(x + 3)$.
• $x^2 - 4x + 4$: This is a perfect square trinomial, which can be factored as $(x - 2)^2$ or $(x - 2)(x - 2)$.

After factoring, our expression looks like this:

$$\frac{(x+2)(x-2)}{x+3} \times \frac{4(x+3)}{(x-2)(x-2)}$$

Factoring polynomials is a critical skill in algebra, and it's essential for simplifying rational expressions. The ability to recognize different factoring patterns, such as the difference of squares and perfect square trinomials, can significantly speed up the simplification process. By breaking down each polynomial into its factors, we expose the common terms that can be canceled out, which is the next crucial step in solving the problem. This step not only simplifies the expression but also makes it easier to see the relationships between the different parts of the expression. Mastering polynomial factorization is not just important for this type of problem but is also fundamental for many other areas of mathematics. Therefore, practicing and becoming proficient in factoring various types of polynomials is highly beneficial for overall mathematical competence.

Step 3: Cancel Common Factors

Next, we look for common factors in the numerator and the denominator that can be canceled out. This is the simplification step where we eliminate terms that appear in both the top and bottom of the fraction.

In our expression, we can cancel:

• $(x + 3)$ in the numerator and denominator.
• One $(x - 2)$ in the numerator and denominator.

After canceling the common factors, we are left with:

$$\frac{(x+2)}{1} \times \frac{4}{(x-2)}$$

Simplifying rational expressions by canceling common factors is akin to reducing a regular fraction to its lowest terms. This step is crucial because it eliminates redundancies and presents the expression in its most concise form. The process involves carefully examining the factored form of the rational expression and identifying terms that appear in both the numerator and denominator. By canceling these terms, we are essentially dividing both the numerator and denominator by the same quantity, which does not change the value of the expression but simplifies its appearance. This skill is fundamental in algebra and is used extensively in various mathematical contexts. It's important to be meticulous when canceling factors to avoid errors and ensure the final expression is indeed in its simplest form. After this step, we are just one step away from the final answer, which involves multiplying the remaining terms together.

Step 4: Multiply the Remaining Terms

Finally, we multiply the remaining terms in the numerator and the denominator to get the simplified expression:

$$\frac{(x+2)}{1} \times \frac{4}{(x-2)} = \frac{4(x+2)}{x-2}$$

So, the quotient in reduced form is $\frac{4(x+2)}{x-2}$.

This final step brings all our previous work together to arrive at the solution. Multiplying the remaining terms is a straightforward process, but it's crucial to ensure accuracy. After canceling out the common factors, we are left with a much simpler expression, which makes the multiplication step easier to manage. The final result, in this case, $\frac{4(x+2)}{x-2}$, represents the simplified form of the original division problem. This means that we have successfully divided the given rational expressions and expressed the result in its most reduced form. Understanding the steps involved in this process, from rewriting division as multiplication to factoring polynomials and canceling common factors, is essential for mastering rational expressions. These skills are not only important for algebra but also for more advanced mathematical topics. Therefore, practicing and gaining confidence in these techniques is highly valuable for mathematical proficiency.

Conclusion

By following these steps, we found that the quotient of the given rational expressions in reduced form is $\frac{4(x+2)}{x-2}$. This matches option A. Remember, the key to solving these types of problems is to rewrite division as multiplication, factor the polynomials, cancel common factors, and then multiply the remaining terms. Practice makes perfect, so keep working on these types of problems to build your skills!

For more resources on rational expressions and algebra, you can visit websites like Khan Academy.