Simplify The Rational Expression.$\[ \frac{3x^2 + 2x - 8}{3x^2 - 19x + 20} \div \frac{x^2 - X - 6}{x^2 - 2x - 3} = \square \\]

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 delve into the world of rational expressions and explore the process of simplifying a given rational expression. We will use the expression $\frac{3x^2 + 2x - 8}{3x^2 - 19x + 20} \div \frac{x^2 - x - 6}{x^2 - 2x - 3}$ as a case study and demonstrate the step-by-step process of simplifying it.

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. To simplify a rational expression, we need to find the greatest common factor (GCF) of the numerator and denominator and cancel it out.

Step 1: Factor the Numerator and Denominator

The first step in simplifying the rational expression is to factor the numerator and denominator. We can factor the numerator and denominator as follows:

$\frac{3x^2 + 2x - 8}{3x^2 - 19x + 20} = \frac{(3x - 4)(x + 2)}{(3x - 4)(x - 5)}$

$\frac{x^2 - x - 6}{x^2 - 2x - 3} = \frac{(x - 3)(x + 2)}{(x - 3)(x + 1)}$

Step 2: Cancel Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out common factors. We can see that both the numerator and denominator have a common factor of $(x + 2)$ and $(3x - 4)$. We can cancel these factors out as follows:

$\frac{(3x - 4)(x + 2)}{(3x - 4)(x - 5)} \div \frac{(x - 3)(x + 2)}{(x - 3)(x + 1)} = \frac{x + 2}{x - 5} \div \frac{x + 1}{x - 3}$

Step 3: Simplify the Division

Now that we have canceled out common factors, we can simplify the division. We can simplify the division by multiplying the numerator and denominator by the reciprocal of the divisor. We can multiply the numerator and denominator by $\frac{x - 3}{x - 3}$ as follows:

$\frac{x + 2}{x - 5} \div \frac{x + 1}{x - 3} = \frac{x + 2}{x - 5} \cdot \frac{x - 3}{x + 1}$

Step 4: Simplify the Expression

Now that we have simplified the division, we can simplify the expression. We can simplify the expression by multiplying the numerator and denominator as follows:

$\frac{x + 2}{x - 5} \cdot \frac{x - 3}{x + 1} = \frac{(x + 2)(x - 3)}{(x - 5)(x + 1)}$

Conclusion

In this article, we have demonstrated the step-by-step process of simplifying a rational expression. We have used the expression $\frac{3x^2 + 2x - 8}{3x^2 - 19x + 20} \div \frac{x^2 - x - 6}{x^2 - 2x - 3}$ as a case study and simplified it by canceling out common factors and simplifying the division. We have also provided a general overview of rational expressions and the process of simplifying them.

Final Answer

The final answer to the problem is $\boxed{\frac{(x + 2)(x - 3)}{(x - 5)(x + 1)}}$.

Tips and Tricks

• When simplifying rational expressions, it is essential to factor the numerator and denominator.
• Cancel out common factors in the numerator and denominator.
• Simplify the division by multiplying the numerator and denominator by the reciprocal of the divisor.
• Simplify the expression by multiplying the numerator and denominator.

Common Mistakes

• Failing to factor the numerator and denominator.
• Failing to cancel out common factors.
• Failing to simplify the division.
• Failing to simplify the expression.

Real-World Applications

Rational expressions have numerous real-world applications in fields such as engineering, economics, and physics. They are used to model real-world situations and make predictions about future events. For example, rational expressions can be used to model the growth of a population, the spread of a disease, or the behavior of a physical system.

Conclusion

Introduction

In our previous article, we demonstrated the step-by-step process of simplifying a rational expression. We used the expression $\frac{3x^2 + 2x - 8}{3x^2 - 19x + 20} \div \frac{x^2 - x - 6}{x^2 - 2x - 3}$ as a case study and simplified it by canceling out common factors and simplifying the division. In this article, we will answer some frequently asked questions about simplifying rational expressions.

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: Why is it important to simplify rational expressions?

A: Simplifying rational expressions is important because it helps to:

• Reduce the complexity of the expression
• Make it easier to solve equations and inequalities
• Make it easier to graph functions
• Make it easier to make predictions about future events

Q: How do I simplify a rational expression?

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

1. Factor the numerator and denominator
2. Cancel out common factors
3. Simplify the division
4. Simplify the expression

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

A: The greatest common factor (GCF) is the largest factor that divides both the numerator and denominator of a rational expression.

Q: How do I find the GCF of two expressions?

A: To find the GCF of two expressions, look for the largest factor that divides both expressions. You can use the following steps:

1. List the factors of each expression
2. Identify the common factors
3. Choose the largest common factor

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

A: Simplifying a rational expression involves canceling out common factors and simplifying the division. Reducing a rational expression involves canceling out common factors, but not simplifying the division.

Q: Can I simplify a rational expression with a zero denominator?

A: No, you cannot simplify a rational expression with a zero denominator. A rational expression with a zero denominator is undefined.

Q: Can I simplify a rational expression with a negative denominator?

A: Yes, you can simplify a rational expression with a negative denominator. However, you must be careful when simplifying the expression to avoid introducing extraneous solutions.

Q: How do I know if a rational expression is already simplified?

A: To determine if a rational expression is already simplified, look for the following:

• Is the numerator and denominator already factored?
• Are there any common factors that can be canceled out?
• Is the division already simplified?

If the answer is no, then the rational expression is not already simplified.

Conclusion

In conclusion, simplifying rational expressions is a crucial skill for any math enthusiast. By following the step-by-step process outlined in this article, we can simplify even the most complex rational expressions. We have also answered some frequently asked questions about simplifying rational expressions and provided tips and tricks for simplifying rational expressions.

Final Answer

The final answer to the problem is $\boxed{\frac{(x + 2)(x - 3)}{(x - 5)(x + 1)}}$.

Tips and Tricks

• When simplifying rational expressions, it is essential to factor the numerator and denominator.
• Cancel out common factors in the numerator and denominator.
• Simplify the division by multiplying the numerator and denominator by the reciprocal of the divisor.
• Simplify the expression by multiplying the numerator and denominator.

Common Mistakes

• Failing to factor the numerator and denominator.
• Failing to cancel out common factors.
• Failing to simplify the division.
• Failing to simplify the expression.

Real-World Applications

Rational expressions have numerous real-world applications in fields such as engineering, economics, and physics. They are used to model real-world situations and make predictions about future events. For example, rational expressions can be used to model the growth of a population, the spread of a disease, or the behavior of a physical system.