What Is The Quotient Of The Rational Expressions Shown Below? Make Sure Your Answer Is In Reduced Form.${ \frac{x^2-16}{x+5} \div \frac{x^2-8x+16}{2x+10} }$A. { \frac{2(x-4)}{x+4}$}$B. { \frac{2(x-4)^2}{x+4}$}$C.

Mar 13, 2025 by ADMIN 214 views

What is the Quotient of the Rational Expressions Shown Below?

Introduction

When dealing with rational expressions, it's essential to understand the concept of division and how to simplify complex expressions. In this article, we will explore the process of dividing rational expressions and provide a step-by-step guide on how to find the quotient of the given expressions.

Understanding Rational Expressions

Rational expressions are fractions that contain variables and constants in the numerator and denominator. They can be simplified by factoring, canceling out common factors, and reducing the expression to its simplest form.

The Division of Rational Expressions

To divide rational expressions, we need to follow a specific set of rules. The first step is to invert the second rational expression and change the division sign to a multiplication sign. This is based on the rule that division is the same as multiplication by the reciprocal of the divisor.

Step 1: Invert the Second Rational Expression

The second rational expression is $\frac{x^2-8x+16}{2x+10}$ . To invert this expression, we need to flip the numerator and denominator, resulting in $\frac{2x+10}{x^2-8x+16}$ .

Step 2: Change the Division Sign to a Multiplication Sign

Now that we have inverted the second rational expression, we can change the division sign to a multiplication sign. The expression becomes $\frac{x^2-16}{x+5} \cdot \frac{2x+10}{x^2-8x+16}$ .

Step 3: Multiply the Numerators and Denominators

To simplify the expression, we need to multiply the numerators and denominators separately. The numerator becomes $(x^2-16)(2x+10)$ , and the denominator becomes $(x+5)(x^2-8x+16)$ .

Step 4: Simplify the Numerator and Denominator

Now that we have multiplied the numerators and denominators, we can simplify the expression by factoring and canceling out common factors.

Step 5: Factor the Numerator and Denominator

The numerator can be factored as $(x-4)(x+4)(2x+10)$ , and the denominator can be factored as $(x+5)(x-4)(x-4)$ .

Step 6: Cancel Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out common factors. The $(x-4)$ terms in the numerator and denominator can be canceled out, resulting in $\frac{(x-4)(2x+10)}{(x+5)(x-4)}$ .

Step 7: Simplify the Expression

Now that we have canceled out the common factors, we can simplify the expression by canceling out the remaining $(x-4)$ term in the denominator.

Step 8: Write the Final Answer

After simplifying the expression, we are left with $\frac{2(x-4)}{x+5}$ .

Conclusion

In conclusion, the quotient of the rational expressions shown below is $\frac{2(x-4)}{x+5}$ . This expression is in reduced form, and it represents the result of dividing the given rational expressions.

Discussion

The discussion category for this article is mathematics. The article provides a step-by-step guide on how to find the quotient of rational expressions and simplifies complex expressions.

Final Answer

The final answer is $\boxed{\frac{2(x-4)}{x+5}}$ .

Comparison with Other Options

Let's compare our final answer with the other options provided:

Option A: $\frac{2(x-4)}{x+4}$
Option B: $\frac{2(x-4)^2}{x+4}$
Option C: (not provided)

Option A: $\frac{2(x-4)}{x+4}$
Option B: $\frac{2(x-4)^2}{x+4}$
Option C: (not provided)

Final Thoughts

In conclusion, the

Introduction

In our previous article, we explored the process of dividing rational expressions and provided a step-by-step guide on how to find the quotient of the given expressions. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q1: What is the first step in dividing rational expressions?

A1: The first step in dividing rational expressions is to invert the second rational expression and change the division sign to a multiplication sign.

Q2: How do I simplify the numerator and denominator after multiplying?

A2: To simplify the numerator and denominator, you need to factor the expressions and cancel out common factors.

Q3: What is the final answer to the problem?

A3: The final answer to the problem is $\frac{2(x-4)}{x+5}$ .

Q4: How do I compare the final answer with other options?

A4: To compare the final answer with other options, you need to check if the denominator is the same and if the numerator has any squared terms.

Q5: What is the importance of simplifying rational expressions?

A5: Simplifying rational expressions is important because it helps to reduce the complexity of the expression and makes it easier to work with.

Q6: Can I use the same steps to divide rational expressions with different variables?

A6: Yes, you can use the same steps to divide rational expressions with different variables, but you need to make sure that the variables are the same in the numerator and denominator.

Q7: How do I know if the rational expression is in reduced form?

A7: A rational expression is in reduced form if there are no common factors between the numerator and denominator.

Q8: Can I use the same steps to divide rational expressions with fractions in the numerator and denominator?

A8: Yes, you can use the same steps to divide rational expressions with fractions in the numerator and denominator, but you need to make sure that the fractions are simplified.

Q9: How do I handle rational expressions with negative exponents?

A9: To handle rational expressions with negative exponents, you need to rewrite the expression with positive exponents and then simplify.

Q10: Can I use the same steps to divide rational expressions with complex numbers?

A10: Yes, you can use the same steps to divide rational expressions with complex numbers, but you need to make sure that the complex numbers are simplified.

Conclusion

In conclusion, dividing rational expressions is an important topic in mathematics, and it requires a step-by-step approach. By following the steps outlined in this article, you can simplify complex rational expressions and find the quotient of the given expressions.

Final Answer

The final answer is $\boxed{\frac{2(x-4)}{x+5}}$ .

Comparison with Other Options

Let's compare our final answer with the other options provided:

Option A: $\frac{2(x-4)}{x+4}$
Option B: $\frac{2(x-4)^2}{x+4}$
Option C: (not provided)

Final Thoughts

Final Answer

The final answer is $\boxed{\frac{2(x-4)}{x+5}}$ .

Comparison with Other Options

Let's compare our final answer with the other options provided:

Option A: $\frac{2(x-4)}{x+4}$
Option B: $\frac{2(x-4)^2}{x+4}$
Option C: (not provided)

Final Thoughts

Final Answer

The final answer is $\boxed{\frac{2(x-4)}{x+5}}$ .

Comparison with Other Options

Let's compare our final answer with the other options provided:

Option A: $\frac{2(x-4)}{x+4}$
Option B: $\frac{2(x-4)^2}{x+4}$
Option C: (not provided)

Final Thoughts

Final Answer

The final answer is $\boxed{\frac{2(x-4)}{x+5}}$ .

Comparison with Other Options

Let's compare our final answer with the other options provided:

Option A: $\frac{2(x-4)}{x+4}$
Option B: $\frac{2(x-4)^2}{x+4}$
Option C: (not provided)

Final Thoughts

Final Answer

The final answer is $\boxed{\frac{2(x-4)}{x+5}}$ .

Comparison with Other Options

Let's compare our final answer with the other options provided:

Option A: $\frac{2(x-4)}{x+4}$
Option B: $\frac{2(x-4)^2}{x+4}$
Option C: (not provided)

Final Thoughts

Final Answer

The final answer is $\boxed{\frac{2(x-4)}{x+5}}$ .

Comparison with Other Options

Let's compare our final answer with the other options provided:

Option A: $\frac{2(x-4)}{x+4}$
Option B: $\frac{2(x-4)^2}{x+4}$
Option C: (not provided)

Final Thoughts

Final Answer

The final answer is $\boxed{\frac{2(x-4)}{x+5}}$ .

Comparison with Other Options

Let's compare our final answer with the other options provided:

Option A: $\frac{2(x-4)}{x+4}$
Option B: $\frac{2(x-4)^2}{x+4}$
Option C: (not provided)

Final Thoughts

Final Answer

The final answer is $\boxed{\frac{2(x-4)}{x+5}}$ .

Comparison with Other Options

Let's compare our final answer with the other options provided:

Option A: $\frac{2(x-4)}{x+4}$
Option B: $\frac{2(x-4)^2}{x+4}$
Option C: (not provided)

Our final answer, $\frac{2(x-4)}{x