Which Of The Following Is Equal To The Rational Expression When X ≠ − 2 X \neq -2 X  = − 2 Or 3? X 2 + 5 X + 6 X 2 − X − 6 \frac{x^2 + 5x + 6}{x^2 - X - 6} X 2 − X − 6 X 2 + 5 X + 6 ​ A. X + 3 X − 3 \frac{x+3}{x-3} X − 3 X + 3 ​ B. X + 2 X − 3 \frac{x+2}{x-3} X − 3 X + 2 ​ C. X + 3 X − 2 \frac{x+3}{x-2} X − 2 X + 3 ​ D. X + 2 X − 2 \frac{x+2}{x-2} X − 2 X + 2 ​

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 given problem: $\frac{x^2 + 5x + 6}{x^2 - x - 6}$ when $x \neq -2$ or 3. We will examine the options provided and determine which one is equal to the given rational expression.

Understanding Rational Expressions

A rational expression is a fraction that contains variables and/or constants in the numerator and denominator. To simplify a rational expression, we need to factor both the numerator and denominator, and then cancel out any common factors.

Factoring the Numerator and Denominator

Let's start by factoring the numerator and denominator of the given rational expression:

$\frac{x^2 + 5x + 6}{x^2 - x - 6}$

We can factor the numerator as follows:

$x^2 + 5x + 6 = (x + 3)(x + 2)$

And the denominator can be factored as:

$x^2 - x - 6 = (x - 3)(x + 2)$

Canceling Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out any common factors. In this case, we have a common factor of $(x + 2)$ in both the numerator and denominator. We can cancel this factor out, leaving us with:

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

Evaluating the Options

Now that we have simplified the rational expression, we can evaluate the options provided:

A. $\frac{x+3}{x-3}$ B. $\frac{x+2}{x-3}$ C. $\frac{x+3}{x-2}$ D. $\frac{x+2}{x-2}$

Based on our simplification, we can see that option A is equal to the given rational expression.

Conclusion

In conclusion, the correct answer is option A: $\frac{x+3}{x-3}$. This is because we were able to simplify the rational expression by factoring the numerator and denominator, and then canceling out the common factor of $(x + 2)$. This process allowed us to arrive at the simplified expression, which is equal to option A.

Additional Tips and Tricks

When simplifying rational expressions, it's essential to remember the following tips and tricks:

• Factor the numerator and denominator completely.
• Cancel out any common factors.
• Be careful when canceling out factors, as this can lead to errors.
• Use the distributive property to expand the numerator and denominator if necessary.

By following these tips and tricks, you can simplify rational expressions with ease and arrive at the correct solution.

Real-World Applications

Rational expressions have numerous real-world applications, including:

• Algebraic geometry: Rational expressions are used to describe curves and surfaces in algebraic geometry.
• Calculus: Rational expressions are used to describe functions and their derivatives in calculus.
• Physics: Rational expressions are used to describe physical systems and their behavior in physics.

Introduction

In our previous article, we explored the process of simplifying rational expressions, with a focus on the given problem: $\frac{x^2 + 5x + 6}{x^2 - x - 6}$ when $x \neq -2$ or 3. We also examined the options provided and determined which one is equal to the given rational expression. In this article, we will provide a Q&A guide to help you better understand the process of 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 denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor both the numerator and denominator, and then cancel out any common factors.

Q: What is factoring?

A: Factoring is the process of expressing an algebraic expression as a product of simpler expressions, called factors.

Q: How do I factor a rational expression?

A: To factor a rational expression, you need to factor the numerator and denominator separately, and then cancel out any common factors.

Q: What is a common factor?

A: A common factor is a factor that appears in both the numerator and denominator of a rational expression.

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to divide both the numerator and denominator by the common factor.

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

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

• Not factoring the numerator and denominator completely.
• Not canceling out common factors.
• Canceling out factors that are not common to both the numerator and denominator.

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

A: A rational expression is already simplified if there are no common factors that can be canceled out.

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 canceling out common factors, as this can lead to errors.

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 rewrite the expression with a positive exponent and then simplify.

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 canceling out common factors, as this can lead to errors.

Conclusion

In conclusion, simplifying rational expressions is a crucial skill for any math enthusiast, and understanding the process can help you arrive at the correct solution. By following the tips and tricks outlined in this article, you can simplify rational expressions with ease and apply them to real-world problems.

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

Try simplifying the following rational expressions:

1. $\frac{x^2 + 4x + 4}{x^2 + 2x + 1}$
2. $\frac{x^2 - 4x + 4}{x^2 - 2x + 1}$
3. $\frac{x^2 + 2x + 1}{x^2 + 4x + 4}$

Answer Key

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