Which Of The Following Is The Quotient Of The Rational Expressions Shown Below?$\[ \frac{x-4}{2x^2} \div \frac{2x+3}{x+4} \\]A. \[$\frac{x^2-16}{4x^3+6x^2}\$\]B. \[$\frac{2x}{2x^2+2x+3}\$\]C. \[$\frac{-2x-3}{2x^2}\$\]D.

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Introduction

Rational expressions are a fundamental concept in algebra, and dividing them is a crucial operation that requires a clear understanding of the rules and procedures involved. In this article, we will explore the process of dividing rational expressions, focusing on the quotient of the given expressions. We will delve into the step-by-step process, highlighting the key concepts and techniques required to arrive at the correct solution.

Understanding Rational Expressions

Before we dive into the division process, it's essential to understand what rational expressions are. A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified, added, subtracted, multiplied, and divided, just like regular fractions.

The Division Process

To divide rational expressions, we follow a specific procedure:

  1. Invert and Multiply: When dividing rational expressions, we invert the second expression (i.e., flip the numerator and denominator) and multiply the two expressions.
  2. Simplify the Expression: After inverting and multiplying, we simplify the resulting expression by canceling out any common factors in the numerator and denominator.

Applying the Division Process

Now that we have a clear understanding of the division process, let's apply it to the given rational expressions:

{ \frac{x-4}{2x^2} \div \frac{2x+3}{x+4} \}

Step 1: Invert and Multiply

To divide the given rational expressions, we invert the second expression and multiply the two expressions:

{ \frac{x-4}{2x^2} \cdot \frac{x+4}{2x+3} \}

Step 2: Simplify the Expression

After inverting and multiplying, we simplify the resulting expression by canceling out any common factors in the numerator and denominator:

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

Expanding the Numerator

To simplify the expression further, we expand the numerator:

{ \frac{x^2-16}{2x^2(2x+3)} \}

Simplifying the Expression

Now that we have expanded the numerator, we can simplify the expression by canceling out any common factors in the numerator and denominator:

{ \frac{x^2-16}{2x^2(2x+3)} \}

Final Answer

After simplifying the expression, we arrive at the final answer:

{ \frac{x^2-16}{4x^3+6x^2} \}

Conclusion

In this article, we explored the process of dividing rational expressions, focusing on the quotient of the given expressions. We applied the division process, inverting and multiplying the expressions, and simplifying the resulting expression by canceling out any common factors in the numerator and denominator. The final answer is x2−164x3+6x2\frac{x^2-16}{4x^3+6x^2}.

Comparison with Answer Choices

Now that we have arrived at the final answer, let's compare it with the given answer choices:

A. x2−164x3+6x2\frac{x^2-16}{4x^3+6x^2} B. 2x2x2+2x+3\frac{2x}{2x^2+2x+3} C. −2x−32x2\frac{-2x-3}{2x^2} D. (Not provided)

Based on our calculation, the correct answer is:

A. x2−164x3+6x2\frac{x^2-16}{4x^3+6x^2}

Final Thoughts

Introduction

Dividing rational expressions can be a challenging task, especially for those who are new to algebra. In this article, we will address some of the most frequently asked questions (FAQs) on dividing rational expressions, providing clear and concise answers to help you better understand the concept.

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

A: The first step in dividing rational expressions is to invert the second expression (i.e., flip the numerator and denominator) and multiply the two expressions.

Q: Why do we invert and multiply when dividing rational expressions?

A: Inverting and multiplying is a fundamental concept in algebra that allows us to simplify the division process. By inverting the second expression, we can eliminate the fraction and make the division process easier to manage.

Q: How do I simplify the expression after inverting and multiplying?

A: After inverting and multiplying, you can simplify the expression by canceling out any common factors in the numerator and denominator. This will help you arrive at the final answer.

Q: What is the difference between simplifying and canceling out common factors?

A: Simplifying an expression involves reducing it to its simplest form, while canceling out common factors involves eliminating any common factors in the numerator and denominator.

Q: Can I simplify the expression before inverting and multiplying?

A: No, you should not simplify the expression before inverting and multiplying. This is because simplifying the expression before inverting and multiplying can lead to incorrect results.

Q: What if the expressions have different denominators?

A: If the expressions have different denominators, you can multiply both expressions by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: Can I divide rational expressions with variables in the denominator?

A: Yes, you can divide rational expressions with variables in the denominator. However, you must be careful when simplifying the expression to avoid any errors.

Q: What if the expression has a zero in the denominator?

A: If the expression has a zero in the denominator, you cannot simplify the expression further. In this case, you must leave the expression as is.

Q: Can I use a calculator to divide rational expressions?

A: Yes, you can use a calculator to divide rational expressions. However, it's essential to understand the concept and process involved in dividing rational expressions to ensure accurate results.

Q: How do I know if my answer is correct?

A: To ensure that your answer is correct, you should:

  • Check your work carefully to avoid any errors.
  • Simplify the expression to its simplest form.
  • Verify that the final answer is in the correct format.

Conclusion

Dividing rational expressions can be a challenging task, but with practice and patience, you can become proficient in solving complex problems. By following the step-by-step process outlined in this article and addressing the frequently asked questions, you can better understand the concept and improve your skills in dividing rational expressions.

Additional Resources

For further assistance, you can refer to the following resources:

  • Algebra textbooks and online resources
  • Online tutorials and video lectures
  • Practice problems and worksheets
  • Online communities and forums

Final Thoughts

Dividing rational expressions is a fundamental concept in algebra that requires practice and patience to master. By following the step-by-step process outlined in this article and addressing the frequently asked questions, you can improve your skills and become proficient in solving complex problems. Remember to check your work carefully, simplify the expression to its simplest form, and verify that the final answer is in the correct format.