What Is The Product Of The Rational Expressions Shown Below? Make Sure Your Answer Is In Reduced Form.${ \frac{x-3}{x+7} \cdot \frac{2x}{x-3} }$A. { \frac{2}{x+7}$}$B. { \frac{2}{x-3}$}$C. { \frac{2x}{x-3}$}$D.

Understanding Rational Expressions

Rational expressions are fractions that contain variables and constants in the numerator and denominator. They are used to represent ratios of two algebraic expressions. In this article, we will explore the product of rational expressions and how to simplify them to their reduced form.

The Product of Rational Expressions

The product of rational expressions is found by multiplying the numerators together and the denominators together. This can be represented as:

$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$

where $a$, $b$, $c$, and $d$ are algebraic expressions.

Simplifying Rational Expressions

To simplify 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.

Example: Simplifying a Rational Expression

Let's simplify the rational expression $\frac{x-3}{x+7} \cdot \frac{2x}{x-3}$.

First, we need to multiply the numerators together:

$(x-3) \cdot 2x = 2x^2 - 6x$

Next, we need to multiply the denominators together:

$(x+7) \cdot (x-3) = x^2 + 4x - 21$

Now, we can write the product of the rational expressions as:

$\frac{2x^2 - 6x}{x^2 + 4x - 21}$

Reducing Rational Expressions

To reduce a rational expression, we need to find the GCF of the numerator and denominator. In this case, the GCF is $(x-3)$.

We can factor out the GCF from the numerator and denominator:

$\frac{(x-3)(2x)}{(x+7)(x-3)}$

Now, we can cancel out the common factor $(x-3)$:

$\frac{2x}{x+7}$

Conclusion

In this article, we explored the product of rational expressions and how to simplify them to their reduced form. We used the example of $\frac{x-3}{x+7} \cdot \frac{2x}{x-3}$ to demonstrate how to multiply and simplify rational expressions.

Answer

The product of the rational expressions shown below is $\frac{2x}{x+7}$.

Discussion

This problem requires the student to understand the concept of rational expressions and how to multiply and simplify them. The student needs to be able to identify the GCF of the numerator and denominator and cancel out common factors.

Key Takeaways

• Rational expressions are fractions that contain variables and constants in the numerator and denominator.
• The product of rational expressions is found by multiplying the numerators together and the denominators together.
• To simplify a rational expression, we need to find the GCF of the numerator and denominator.
• To reduce a rational expression, we need to find the GCF of the numerator and denominator and cancel out common factors.

Practice Problems

1. Simplify the rational expression $\frac{x+5}{x-3} \cdot \frac{2x}{x+5}$.
2. Reduce the rational expression $\frac{2x^2 - 6x}{x^2 + 4x - 21}$.
3. Multiply the rational expressions $\frac{x-3}{x+7}$ and $\frac{2x}{x-3}$.

Answer Key

1. $\frac{2x}{x-3}$
2. $\frac{2x}{x+7}$
3. $\frac{2x}{x+7}$
   Frequently Asked Questions (FAQs) About Rational Expressions ================================================================

Q: What is a rational expression?

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

Q: How do I multiply rational expressions?

A: To multiply rational expressions, you need to multiply the numerators together and the denominators together. This can be represented as:

$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you 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.

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

A: The GCF 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 GCF of the numerator and denominator and cancel out common factors.

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

A: Simplifying a rational expression involves finding the GCF of the numerator and denominator, while reducing a rational expression involves canceling out common factors.

Q: Can I simplify or reduce a rational expression if there are no common factors?

A: Yes, you can still simplify or reduce a rational expression even if there are no common factors. In this case, the rational expression is already in its simplest form.

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

A: A rational expression is in its simplest form if there are no common factors between the numerator and denominator.

Q: Can I multiply rational expressions with different variables?

A: Yes, you can multiply rational expressions with different variables. However, you need to make sure that the variables are compatible (i.e., they have the same units).

Q: Can I simplify or reduce a rational expression with different variables?

A: Yes, you can simplify or reduce a rational expression with different variables. However, you need to make sure that the variables are compatible (i.e., they have the same units).

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

A: Some common mistakes to avoid when working with rational expressions include:

• Not simplifying or reducing the rational expression
• Not canceling out common factors
• Not checking for compatibility between variables
• Not following the order of operations

Q: How can I practice working with rational expressions?

A: You can practice working with rational expressions by:

• Simplifying and reducing rational expressions
• Multiplying rational expressions
• Solving equations and inequalities involving rational expressions
• Working with rational expressions in real-world applications

Conclusion

In this article, we have covered some of the most frequently asked questions about rational expressions. We have discussed how to multiply, simplify, and reduce rational expressions, as well as how to avoid common mistakes. By practicing working with rational expressions, you can become more confident and proficient in your ability to work with these important mathematical concepts.

Practice Problems

1. Simplify the rational expression $\frac{x+5}{x-3} \cdot \frac{2x}{x+5}$.
2. Reduce the rational expression $\frac{2x^2 - 6x}{x^2 + 4x - 21}$.
3. Multiply the rational expressions $\frac{x-3}{x+7}$ and $\frac{2x}{x-3}$.

Answer Key

1. $\frac{2x}{x-3}$
2. $\frac{2x}{x+7}$
3. $\frac{2x}{x+7}$