What Is The Product Of The Rational Expressions Shown Below? Make Sure Your Answer Is In Reduced Form. X + 2 X − 5 ⋅ 8 X X + 2 \frac{x+2}{x-5} \cdot \frac{8x}{x+2} X − 5 X + 2 ​ ⋅ X + 2 8 X ​ A. 8 X X − 5 \frac{8x}{x-5} X − 5 8 X ​ B. 8 X − 5 \frac{8}{x-5} X − 5 8 ​ C. 8 X + 2 \frac{8}{x+2} X + 2 8 ​ D.

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Understanding Rational Expressions

Rational expressions are fractions that contain variables and constants in the numerator and denominator. They are used to represent mathematical relationships and can be simplified, added, subtracted, multiplied, and divided. In this article, we will focus on finding the product of rational expressions.

The Product of Rational Expressions

To find the product of rational expressions, we need to multiply the numerators and denominators separately. The product of two rational expressions is a new rational expression where the numerator is the product of the numerators and the denominator is the product of the denominators.

Example: Finding the Product of Rational Expressions

Let's consider the following rational expressions:

x+2x−5⋅8xx+2\frac{x+2}{x-5} \cdot \frac{8x}{x+2}

To find the product, we need to multiply the numerators and denominators separately.

Step 1: Multiply the Numerators

The numerators are x+2x+2 and 8x8x. To multiply them, we need to multiply each term in the first numerator by each term in the second numerator.

(x+2)⋅8x=8x2+16x(x+2) \cdot 8x = 8x^2 + 16x

Step 2: Multiply the Denominators

The denominators are x−5x-5 and x+2x+2. To multiply them, we need to multiply each term in the first denominator by each term in the second denominator.

(x−5)⋅(x+2)=x2−5x+2x−10(x-5) \cdot (x+2) = x^2 - 5x + 2x - 10

Simplifying the expression, we get:

x2−3x−10x^2 - 3x - 10

Step 3: Write the Product of Rational Expressions

Now that we have multiplied the numerators and denominators, we can write the product of rational expressions.

8x2+16xx2−3x−10\frac{8x^2 + 16x}{x^2 - 3x - 10}

Reducing the Product of Rational Expressions

To reduce the product of rational expressions, we need to find the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest expression that divides both the numerator and denominator without leaving a remainder.

Finding the GCF

The GCF of 8x2+16x8x^2 + 16x and x2−3x−10x^2 - 3x - 10 is xx.

Reducing the Product of Rational Expressions

Now that we have found the GCF, we can reduce the product of rational expressions by dividing both the numerator and denominator by the GCF.

8x2+16xx2−3x−10=x(8x+16)x(x−5)−2(x−5)\frac{8x^2 + 16x}{x^2 - 3x - 10} = \frac{x(8x + 16)}{x(x - 5) - 2(x - 5)}

Simplifying the expression, we get:

8x(x+2)(x−5)(x−2)\frac{8x(x + 2)}{(x - 5)(x - 2)}

Conclusion

In this article, we have learned how to find the product of rational expressions. We have also learned how to reduce the product of rational expressions by finding the greatest common factor (GCF) of the numerator and denominator. The product of rational expressions is a new rational expression where the numerator is the product of the numerators and the denominator is the product of the denominators.

Answer

The product of the rational expressions shown below is:

8xx−5\frac{8x}{x-5}

This is the correct answer because we have reduced the product of rational expressions by finding the GCF of the numerator and denominator.

Discussion

This article has provided a step-by-step guide on how to find the product of rational expressions. It has also provided examples and explanations to help readers understand the concept. The article has also discussed the importance of reducing rational expressions to their simplest form.

Key Takeaways

  • Rational expressions are fractions that contain variables and constants in the numerator and denominator.
  • The product of two rational expressions is a new rational expression where the numerator is the product of the numerators and the denominator is the product of the denominators.
  • To reduce a rational expression, we need to find the greatest common factor (GCF) of the numerator and denominator.
  • The GCF is the largest expression that divides both the numerator and denominator without leaving a remainder.

Final Thoughts

Frequently Asked Questions

In this article, we will answer some frequently asked questions about the product of rational expressions.

Q: What is the product of rational expressions?

A: The product of rational expressions is a new rational expression where the numerator is the product of the numerators and the denominator is the product of the denominators.

Q: How do I find the product of rational expressions?

A: To find the product of rational expressions, you need to multiply the numerators and denominators separately. The product of two rational expressions is a new rational expression where the numerator is the product of the numerators and the denominator is the product of the denominators.

Q: What is the greatest common factor (GCF) of rational expressions?

A: The greatest common factor (GCF) of rational expressions is the largest expression that divides both the numerator and denominator without leaving a remainder.

Q: How do I reduce a rational expression?

A: To reduce a rational expression, you need to find the greatest common factor (GCF) of the numerator and denominator. Then, you can divide both the numerator and denominator by the GCF to simplify the expression.

Q: What is the importance of reducing rational expressions?

A: Reducing rational expressions is important because it helps to simplify the expression and make it easier to work with. It also helps to eliminate any common factors that may be present in the numerator and denominator.

Q: Can I add or subtract rational expressions?

A: Yes, you can add or subtract rational expressions. However, you need to make sure that the denominators are the same before you can add or subtract the numerators.

Q: How do I multiply rational expressions with different denominators?

A: To multiply rational expressions with different denominators, you need to multiply the numerators and denominators separately. Then, you can simplify the expression by finding the greatest common factor (GCF) of the numerator and denominator.

Q: Can I divide rational expressions?

A: Yes, you can divide rational expressions. However, you need to make sure that the denominator is not equal to zero before you can divide the numerator by the denominator.

Q: What is the product of rational expressions with variables in the denominator?

A: The product of rational expressions with variables in the denominator is a new rational expression where the numerator is the product of the numerators and the denominator is the product of the denominators.

Q: How do I simplify rational expressions with variables in the denominator?

A: To simplify rational expressions with variables in the denominator, you need to find the greatest common factor (GCF) of the numerator and denominator. Then, you can divide both the numerator and denominator by the GCF to simplify the expression.

Q: Can I use rational expressions in real-world problems?

A: Yes, you can use rational expressions in real-world problems. Rational expressions are used to represent mathematical relationships and can be simplified, added, subtracted, multiplied, and divided.

Q: What are some common applications of rational expressions?

A: Some common applications of rational expressions include:

  • Algebra
  • Geometry
  • Trigonometry
  • Calculus
  • Physics
  • Engineering

Conclusion

In this article, we have answered some frequently asked questions about the product of rational expressions. We have also discussed the importance of reducing rational expressions and how to simplify rational expressions with variables in the denominator. Rational expressions are used to represent mathematical relationships and can be simplified, added, subtracted, multiplied, and divided. By understanding how to work with rational expressions, you can solve mathematical problems and equations with ease.

Key Takeaways

  • The product of rational expressions is a new rational expression where the numerator is the product of the numerators and the denominator is the product of the denominators.
  • To find the product of rational expressions, you need to multiply the numerators and denominators separately.
  • The greatest common factor (GCF) of rational expressions is the largest expression that divides both the numerator and denominator without leaving a remainder.
  • Reducing rational expressions is important because it helps to simplify the expression and make it easier to work with.
  • Rational expressions can be used in real-world problems and have many common applications.