Which Of The Following Is The Quotient Of The Rational Expressions Shown Below? Ensure Your Answer Is In Reduced Form. 2 X − 1 X + 1 ÷ 3 X 2 X 2 + X \frac{2x-1}{x+1} \div \frac{3x^2}{x^2+x} X + 1 2 X − 1 ​ ÷ X 2 + X 3 X 2 ​ A. 6 X 5 − 3 X 2 X 3 + 2 X 2 + X \frac{6x^5-3x^2}{x^3+2x^2+x} X 3 + 2 X 2 + X 6 X 5 − 3 X 2 ​ B. 2 X − 1 3 X \frac{2x-1}{3x} 3 X 2 X − 1 ​ C.

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Understanding Rational Expression Division

When dealing with rational expressions, division is a crucial operation that requires careful consideration. In this article, we will explore the process of dividing rational expressions and provide a step-by-step guide on how to simplify the quotient. We will also examine a specific problem and determine the correct answer.

The Process of Rational Expression Division

To divide two rational expressions, we need to follow a specific procedure. The process involves inverting the second rational expression and then multiplying it by the first rational expression. This is based on the rule that division is the same as multiplying by the reciprocal of the divisor.

Step 1: Invert the Second Rational Expression

The second rational expression is 3x2x2+x\frac{3x^2}{x^2+x}. To invert it, we need to flip the numerator and denominator, resulting in x2+x3x2\frac{x^2+x}{3x^2}.

Step 2: Multiply the First Rational Expression by the Inverted Second Rational Expression

Now, we need to multiply the first rational expression 2x−1x+1\frac{2x-1}{x+1} by the inverted second rational expression x2+x3x2\frac{x^2+x}{3x^2}. This involves multiplying the numerators and denominators separately.

Step 3: Simplify the Resulting Expression

After multiplying the numerators and denominators, we need to simplify the resulting expression. This involves canceling out any common factors between the numerator and denominator.

Solving the Given Problem

Now, let's apply the process of rational expression division to the given problem: 2x−1x+1÷3x2x2+x\frac{2x-1}{x+1} \div \frac{3x^2}{x^2+x}.

Step 1: Invert the Second Rational Expression

The second rational expression is 3x2x2+x\frac{3x^2}{x^2+x}. To invert it, we need to flip the numerator and denominator, resulting in x2+x3x2\frac{x^2+x}{3x^2}.

Step 2: Multiply the First Rational Expression by the Inverted Second Rational Expression

Now, we need to multiply the first rational expression 2x−1x+1\frac{2x-1}{x+1} by the inverted second rational expression x2+x3x2\frac{x^2+x}{3x^2}. This involves multiplying the numerators and denominators separately.

2x−1x+1⋅x2+x3x2=(2x−1)(x2+x)(x+1)(3x2)\frac{2x-1}{x+1} \cdot \frac{x^2+x}{3x^2} = \frac{(2x-1)(x^2+x)}{(x+1)(3x^2)}

Step 3: Simplify the Resulting Expression

After multiplying the numerators and denominators, we need to simplify the resulting expression. This involves canceling out any common factors between the numerator and denominator.

(2x−1)(x2+x)(x+1)(3x2)=2x3+2x2−x2−x3x3+3x2\frac{(2x-1)(x^2+x)}{(x+1)(3x^2)} = \frac{2x^3+2x^2-x^2-x}{3x^3+3x^2}

=2x3+x2−x3x3+3x2= \frac{2x^3+x^2-x}{3x^3+3x^2}

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

=2x2+x−13(x2+x)= \frac{2x^2+x-1}{3(x^2+x)}

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

=2x−13= \frac{2x-1}{3}

Conclusion

In conclusion, the quotient of the rational expressions 2x−1x+1÷3x2x2+x\frac{2x-1}{x+1} \div \frac{3x^2}{x^2+x} is 2x−13\frac{2x-1}{3}. This is the reduced form of the quotient, and it is the correct answer to the given problem.

Final Answer

The final answer is 2x−13\boxed{\frac{2x-1}{3}}.

Understanding Rational Expression Division

In our previous article, we explored the process of dividing rational expressions and provided a step-by-step guide on how to simplify the quotient. However, we understand that there may be some questions and concerns that you may have. In this article, we will address some of the most frequently asked questions about rational expression division.

Q: What is the rule for dividing rational expressions?

A: The rule for dividing rational expressions is to invert the second rational expression and then multiply it by the first rational expression. This is based on the rule that division is the same as multiplying by the reciprocal of the divisor.

Q: How do I invert a rational expression?

A: To invert a rational expression, you need to flip the numerator and denominator. For example, if you have the rational expression ab\frac{a}{b}, the inverted rational expression would be ba\frac{b}{a}.

Q: What is the difference between multiplying and dividing rational expressions?

A: Multiplying and dividing rational expressions are two different operations. When you multiply rational expressions, you multiply the numerators and denominators separately. When you divide rational expressions, you invert the second rational expression and then multiply it by the first rational expression.

Q: How do I simplify a rational expression after dividing?

A: After dividing rational expressions, you need to simplify the resulting expression by canceling out any common factors between the numerator and denominator.

Q: What are some common mistakes to avoid when dividing rational expressions?

A: Some common mistakes to avoid when dividing rational expressions include:

  • Not inverting the second rational expression
  • Not multiplying the numerators and denominators separately
  • Not simplifying the resulting expression
  • Not canceling out common factors between the numerator and denominator

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

A: Yes, you can use a calculator to divide rational expressions. However, it's always a good idea to check your work by simplifying the resulting expression manually.

Q: How do I know if a rational expression is in its simplest form?

A: A rational expression is in its simplest form when there are no common factors between the numerator and denominator. You can check this by factoring the numerator and denominator and canceling out any common factors.

Q: Can I divide rational expressions with different signs?

A: Yes, you can divide rational expressions with different signs. When you divide rational expressions with different signs, you need to follow the rules of division, including inverting the second rational expression and multiplying it by the first rational expression.

Q: What are some real-world applications of rational expression division?

A: Rational expression division has many real-world applications, including:

  • Calculating rates and ratios
  • Finding the area and perimeter of shapes
  • Solving problems involving motion and velocity
  • Analyzing data and statistics

Conclusion

In conclusion, rational expression division is an important concept in algebra that requires careful attention to detail. By following the rules of division and simplifying the resulting expression, you can solve problems involving rational expressions with ease. We hope that this article has helped to clarify any questions or concerns you may have had about rational expression division.

Final Answer

The final answer is 2x−13\boxed{\frac{2x-1}{3}}.