Which Of The Following Is The Product Of The Rational Expressions Shown Here? 1 X + 2 ⋅ X X − 2 \frac{1}{x+2} \cdot \frac{x}{x-2} X + 2 1 ​ ⋅ X − 2 X ​ A. X + 1 2 X \frac{x+1}{2x} 2 X X + 1 ​ B. X + 1 X 2 − 4 \frac{x+1}{x^2-4} X 2 − 4 X + 1 ​ C. X X 2 − 4 \frac{x}{x^2-4} X 2 − 4 X ​

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Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore the process of simplifying rational expressions, with a focus on the product of two rational expressions. We will use the given example to demonstrate the step-by-step process of simplifying rational expressions.

Understanding Rational Expressions

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by canceling out common factors in the numerator and denominator. The product of two rational expressions is obtained by multiplying the numerators and denominators separately.

The Product of Rational Expressions

The given problem is to find the product of the rational expressions 1x+2\frac{1}{x+2} and xx−2\frac{x}{x-2}. To simplify this expression, we will follow the steps outlined below:

Step 1: Multiply the Numerators and Denominators

To find the product of the rational expressions, we multiply the numerators and denominators separately.

1x+2⋅xx−2=1⋅x(x+2)⋅(x−2)\frac{1}{x+2} \cdot \frac{x}{x-2} = \frac{1 \cdot x}{(x+2) \cdot (x-2)}

Step 2: Simplify the Numerator and Denominator

Now, we simplify the numerator and denominator separately.

x(x+2)⋅(x−2)\frac{x}{(x+2) \cdot (x-2)}

Step 3: Factor the Denominator

The denominator can be factored as a difference of squares.

(x+2)⋅(x−2)=x2−4(x+2) \cdot (x-2) = x^2 - 4

Step 4: Write the Final Answer

Now, we can write the final answer by combining the simplified numerator and denominator.

xx2−4\frac{x}{x^2 - 4}

Conclusion

In this article, we have demonstrated the step-by-step process of simplifying rational expressions, with a focus on the product of two rational expressions. We have used the given example to illustrate the process of multiplying the numerators and denominators, simplifying the numerator and denominator, factoring the denominator, and writing the final answer. By following these steps, you can simplify rational expressions with ease.

Answer Options

Based on the simplified expression, we can conclude that the correct answer is:

C. xx2−4\frac{x}{x^2-4}

Additional Examples

To reinforce your understanding of simplifying rational expressions, here are some additional examples:

  • 2x+1⋅xx−1=2xx2−1\frac{2}{x+1} \cdot \frac{x}{x-1} = \frac{2x}{x^2 - 1}
  • 3x−2⋅xx+2=3xx2−4\frac{3}{x-2} \cdot \frac{x}{x+2} = \frac{3x}{x^2 - 4}
  • 4x+3⋅xx−3=4xx2−9\frac{4}{x+3} \cdot \frac{x}{x-3} = \frac{4x}{x^2 - 9}

Tips and Tricks

Here are some tips and tricks to help you simplify rational expressions:

  • Always multiply the numerators and denominators separately.
  • Simplify the numerator and denominator separately.
  • Factor the denominator whenever possible.
  • Write the final answer by combining the simplified numerator and denominator.

Introduction

In our previous article, we explored the process of simplifying rational expressions, with a focus on the product of two rational expressions. In this article, we will answer some frequently asked questions about simplifying rational expressions.

Q&A

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow these steps:

  1. Multiply the numerators and denominators separately.
  2. Simplify the numerator and denominator separately.
  3. Factor the denominator whenever possible.
  4. Write the final answer by combining the simplified numerator and denominator.

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 constants in the numerator and/or denominator.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator. However, you need to be careful when simplifying the expression, as the variable may cancel out.

Q: How do I simplify a rational expression with a negative exponent?

A: To simplify a rational expression with a negative exponent, you need to follow these steps:

  1. Rewrite the expression with a positive exponent.
  2. Simplify the numerator and denominator separately.
  3. Factor the denominator whenever possible.
  4. Write the final answer by combining the simplified numerator and denominator.

Q: Can I simplify a rational expression with a fraction in the denominator?

A: Yes, you can simplify a rational expression with a fraction in the denominator. However, you need to be careful when simplifying the expression, as the fraction may cancel out.

Q: How do I simplify a rational expression with a variable in the numerator and denominator?

A: To simplify a rational expression with a variable in the numerator and denominator, you need to follow these steps:

  1. Factor the numerator and denominator separately.
  2. Cancel out any common factors.
  3. Simplify the remaining expression.

Q: Can I simplify a rational expression with a polynomial in the numerator and denominator?

A: Yes, you can simplify a rational expression with a polynomial in the numerator and denominator. However, you need to be careful when simplifying the expression, as the polynomial may cancel out.

Q: How do I simplify a rational expression with a radical in the numerator and denominator?

A: To simplify a rational expression with a radical in the numerator and denominator, you need to follow these steps:

  1. Simplify the radical in the numerator and denominator separately.
  2. Factor the denominator whenever possible.
  3. Write the final answer by combining the simplified numerator and denominator.

Conclusion

In this article, we have answered some frequently asked questions about simplifying rational expressions. We have covered topics such as the definition of a rational expression, simplifying rational expressions with variables, negative exponents, fractions, polynomials, and radicals. By following the steps outlined in this article, you can simplify rational expressions with ease and become a math whiz!

Additional Resources

For more information on simplifying rational expressions, check out the following resources:

  • Khan Academy: Simplifying Rational Expressions
  • Mathway: Simplifying Rational Expressions
  • Wolfram Alpha: Simplifying Rational Expressions

Practice Problems

To practice simplifying rational expressions, try the following problems:

  • 2x+1⋅xx−1=2xx2−1\frac{2}{x+1} \cdot \frac{x}{x-1} = \frac{2x}{x^2 - 1}
  • 3x−2⋅xx+2=3xx2−4\frac{3}{x-2} \cdot \frac{x}{x+2} = \frac{3x}{x^2 - 4}
  • 4x+3⋅xx−3=4xx2−9\frac{4}{x+3} \cdot \frac{x}{x-3} = \frac{4x}{x^2 - 9}

By practicing these problems, you can become more confident in your ability to simplify rational expressions.