Which Of The Following Is The Quotient Of The Rational Expressions Shown Below?$\frac{x}{3x-1} \div \frac{x-2}{2x}$A. $\frac{4x^2}{6x^2-2x}$ B. $\frac{2x^2}{3x^2-7x+2}$ C. $\frac{3x}{4x-3}$ D.

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Which of the following is the quotient of the rational expressions shown below?

Understanding Rational Expressions and Division

Rational expressions are fractions that contain variables and/or numbers in the numerator and/or denominator. When we divide one rational expression by another, we are essentially multiplying the first expression by the reciprocal of the second expression. This process is crucial in simplifying complex rational expressions and solving equations.

The Quotient of Rational Expressions

To find the quotient of two rational expressions, we follow a simple step-by-step process:

  1. Flip the second rational expression: We take the reciprocal of the second rational expression, which means we flip the numerator and the denominator.
  2. Multiply the first rational expression by the flipped second rational expression: We multiply the numerators together and the denominators together.
  3. Simplify the resulting expression: We simplify the resulting expression by canceling out any common factors in the numerator and the denominator.

Applying the Quotient Formula to the Given Rational Expressions

Let's apply the quotient formula to the given rational expressions:

x3x−1÷x−22x\frac{x}{3x-1} \div \frac{x-2}{2x}

We start by flipping the second rational expression:

x−22x→2xx−2\frac{x-2}{2x} \rightarrow \frac{2x}{x-2}

Next, we multiply the first rational expression by the flipped second rational expression:

x3x−1⋅2xx−2\frac{x}{3x-1} \cdot \frac{2x}{x-2}

We multiply the numerators together and the denominators together:

x⋅2x3x−1⋅x−2\frac{x \cdot 2x}{3x-1 \cdot x-2}

Simplifying the resulting expression, we get:

2x23x2−3x−2\frac{2x^2}{3x^2-3x-2}

Evaluating the Answer Choices

Now that we have the quotient of the rational expressions, let's evaluate the answer choices:

A. 4x26x2−2x\frac{4x^2}{6x^2-2x}

B. 2x23x2−7x+2\frac{2x^2}{3x^2-7x+2}

C. 3x4x−3\frac{3x}{4x-3}

D.

Comparing our result with the answer choices, we can see that:

  • A. 4x26x2−2x\frac{4x^2}{6x^2-2x} is not equal to our result.
  • B. 2x23x2−7x+2\frac{2x^2}{3x^2-7x+2} is not equal to our result.
  • C. 3x4x−3\frac{3x}{4x-3} is not equal to our result.
  • D. is not provided.

However, we can simplify our result further by factoring the denominator:

2x23x2−3x−2\frac{2x^2}{3x^2-3x-2}

Factoring the denominator, we get:

2x23x(x−1)−2(x−1)\frac{2x^2}{3x(x-1)-2(x-1)}

Simplifying further, we get:

2x2(3x−2)(x−1)\frac{2x^2}{(3x-2)(x-1)}

Now, let's compare our result with the answer choices again:

  • A. 4x26x2−2x\frac{4x^2}{6x^2-2x} is not equal to our result.
  • B. 2x23x2−7x+2\frac{2x^2}{3x^2-7x+2} is not equal to our result.
  • C. 3x4x−3\frac{3x}{4x-3} is not equal to our result.
  • D. is not provided.

However, we can see that our result is close to answer choice B. 2x23x2−7x+2\frac{2x^2}{3x^2-7x+2}, but it is not exactly the same.

Conclusion

In conclusion, the quotient of the rational expressions x3x−1÷x−22x\frac{x}{3x-1} \div \frac{x-2}{2x} is 2x23x2−3x−2\frac{2x^2}{3x^2-3x-2}, which can be simplified further to 2x2(3x−2)(x−1)\frac{2x^2}{(3x-2)(x-1)}. This result is not exactly the same as any of the answer choices, but it is close to answer choice B. 2x23x2−7x+2\frac{2x^2}{3x^2-7x+2}.

Final Answer

The final answer is 2x2(3x−2)(x−1)\boxed{\frac{2x^2}{(3x-2)(x-1)}}.
Quotient of Rational Expressions: Frequently Asked Questions

Understanding Rational Expressions and Division

Rational expressions are fractions that contain variables and/or numbers in the numerator and/or denominator. When we divide one rational expression by another, we are essentially multiplying the first expression by the reciprocal of the second expression. This process is crucial in simplifying complex rational expressions and solving equations.

Q: What is the quotient of two rational expressions?

A: The quotient of two rational expressions is the result of dividing one rational expression by another. To find the quotient, we follow a simple step-by-step process:

  1. Flip the second rational expression: We take the reciprocal of the second rational expression, which means we flip the numerator and the denominator.
  2. Multiply the first rational expression by the flipped second rational expression: We multiply the numerators together and the denominators together.
  3. Simplify the resulting expression: We simplify the resulting expression by canceling out any common factors in the numerator and the denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, we follow these steps:

  1. Factor the numerator and the denominator: We factor the numerator and the denominator to identify any common factors.
  2. Cancel out common factors: We cancel out any common factors in the numerator and the denominator.
  3. Simplify the resulting expression: We simplify the resulting expression by multiplying the remaining factors together.

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. A rational expression, on the other hand, is a fraction that contains variables and/or numbers in the numerator and/or denominator.

Q: How do I divide a rational expression by a rational number?

A: To divide a rational expression by a rational number, we follow these steps:

  1. Flip the rational number: We take the reciprocal of the rational number, which means we flip the numerator and the denominator.
  2. Multiply the rational expression by the flipped rational number: We multiply the numerator and the denominator of the rational expression by the flipped rational number.
  3. Simplify the resulting expression: We simplify the resulting expression by canceling out any common factors in the numerator and the denominator.

Q: What is the quotient of the rational expressions x3x−1÷x−22x\frac{x}{3x-1} \div \frac{x-2}{2x}?

A: The quotient of the rational expressions x3x−1÷x−22x\frac{x}{3x-1} \div \frac{x-2}{2x} is 2x23x2−3x−2\frac{2x^2}{3x^2-3x-2}, which can be simplified further to 2x2(3x−2)(x−1)\frac{2x^2}{(3x-2)(x-1)}.

Q: How do I evaluate the answer choices for the quotient of the rational expressions?

A: To evaluate the answer choices for the quotient of the rational expressions, we compare our result with the answer choices. In this case, our result is 2x2(3x−2)(x−1)\frac{2x^2}{(3x-2)(x-1)}, which is not exactly the same as any of the answer choices.

Conclusion

In conclusion, the quotient of two rational expressions is the result of dividing one rational expression by another. To find the quotient, we follow a simple step-by-step process: flip the second rational expression, multiply the first rational expression by the flipped second rational expression, and simplify the resulting expression. We can also simplify a rational expression by factoring the numerator and the denominator and canceling out common factors. By understanding rational expressions and division, we can solve complex equations and simplify rational expressions.

Final Answer

The final answer is 2x2(3x−2)(x−1)\boxed{\frac{2x^2}{(3x-2)(x-1)}}.