Simplify The Expression:${ \frac{2x^2 + 5x + 2}{x^2 + 1x - 6} \div \frac{x^2 + 6x + 8}{2x^2 - 5x + 2} }$

Introduction

When it comes to simplifying complex expressions, one of the most challenging tasks is dividing rational expressions. Rational expressions are fractions that contain variables and constants in the numerator and denominator. Dividing these expressions can be a daunting task, but with the right approach, it can be simplified to a manageable form. In this article, we will explore the steps involved in simplifying the expression $\frac{2x^2 + 5x + 2}{x^2 + 1x - 6} \div \frac{x^2 + 6x + 8}{2x^2 - 5x + 2}$.

Understanding Rational Expressions

Before we dive into simplifying the expression, it's essential to understand what rational expressions are. A rational expression is a fraction that contains variables and constants in the numerator and denominator. Rational expressions can be added, subtracted, multiplied, and divided, just like regular fractions. However, when it comes to dividing rational expressions, we need to follow a specific set of rules to simplify the expression.

The Rules for Dividing Rational Expressions

When dividing rational expressions, we need to follow the following rules:

1. Invert and Multiply: When dividing rational expressions, we need to invert the second fraction and multiply it with the first fraction.
2. Simplify the Expression: After inverting and multiplying, we need to simplify the expression by canceling out any common factors in the numerator and denominator.
3. Check for Any Common Factors: Before simplifying the expression, we need to check if there are any common factors in the numerator and denominator that can be canceled out.

Simplifying the Expression

Now that we have a good understanding of rational expressions and the rules for dividing them, let's simplify the expression $\frac{2x^2 + 5x + 2}{x^2 + 1x - 6} \div \frac{x^2 + 6x + 8}{2x^2 - 5x + 2}$.

Step 1: Invert and Multiply

To simplify the expression, we need to invert the second fraction and multiply it with the first fraction. This means that we need to change the division sign to a multiplication sign and change the second fraction to its reciprocal.

$\frac{2x^2 + 5x + 2}{x^2 + 1x - 6} \div \frac{x^2 + 6x + 8}{2x^2 - 5x + 2} = \frac{2x^2 + 5x + 2}{x^2 + 1x - 6} \times \frac{2x^2 - 5x + 2}{x^2 + 6x + 8}$

Step 2: Simplify the Expression

Now that we have inverted and multiplied the fractions, we need to simplify the expression by canceling out any common factors in the numerator and denominator.

$\frac{2x^2 + 5x + 2}{x^2 + 1x - 6} \times \frac{2x^2 - 5x + 2}{x^2 + 6x + 8} = \frac{(2x^2 + 5x + 2)(2x^2 - 5x + 2)}{(x^2 + 1x - 6)(x^2 + 6x + 8)}$

Step 3: Check for Any Common Factors

Before simplifying the expression, we need to check if there are any common factors in the numerator and denominator that can be canceled out.

$\frac{(2x^2 + 5x + 2)(2x^2 - 5x + 2)}{(x^2 + 1x - 6)(x^2 + 6x + 8)}$

After checking, we find that there are no common factors in the numerator and denominator that can be canceled out.

Step 4: Simplify the Expression

Now that we have checked for any common factors, we can simplify the expression by multiplying the numerators and denominators.

$\frac{(2x^2 + 5x + 2)(2x^2 - 5x + 2)}{(x^2 + 1x - 6)(x^2 + 6x + 8)} = \frac{4x^4 - 25x^3 + 20x^2 + 10x^3 - 25x^2 + 20x + 4x^2 - 10x + 4}{x^4 + 7x^3 - 6x^2 + 6x^3 + 37x^2 + 48x}$

Step 5: Combine Like Terms

Now that we have multiplied the numerators and denominators, we need to combine like terms in the numerator and denominator.

$\frac{4x^4 - 25x^3 + 20x^2 + 10x^3 - 25x^2 + 20x + 4x^2 - 10x + 4}{x^4 + 7x^3 - 6x^2 + 6x^3 + 37x^2 + 48x} = \frac{4x^4 - 15x^3 - 5x^2 + 10x + 4}{x^4 + 13x^3 + 31x^2 + 48x}$

Step 6: Factor the Numerator and Denominator

Now that we have combined like terms, we need to factor the numerator and denominator.

$\frac{4x^4 - 15x^3 - 5x^2 + 10x + 4}{x^4 + 13x^3 + 31x^2 + 48x} = \frac{(2x^2 - 5x + 2)(2x^2 - 5x + 2)}{(x^2 + 6x + 8)(x^2 + 7x + 6)}$

Step 7: Cancel Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out any common factors.

$\frac{(2x^2 - 5x + 2)(2x^2 - 5x + 2)}{(x^2 + 6x + 8)(x^2 + 7x + 6)} = \frac{2x^2 - 5x + 2}{x^2 + 6x + 8}$

Conclusion

In conclusion, simplifying the expression $\frac{2x^2 + 5x + 2}{x^2 + 1x - 6} \div \frac{x^2 + 6x + 8}{2x^2 - 5x + 2}$ involves following a series of steps, including inverting and multiplying the fractions, simplifying the expression, checking for any common factors, and canceling out common factors. By following these steps, we can simplify the expression to its simplest form.

Final Answer

The final answer is $\boxed{\frac{2x^2 - 5x + 2}{x^2 + 6x + 8}}$.

Introduction

In our previous article, we explored the steps involved in simplifying the expression $\frac{2x^2 + 5x + 2}{x^2 + 1x - 6} \div \frac{x^2 + 6x + 8}{2x^2 - 5x + 2}$. In this article, we will answer some of the most frequently asked questions about simplifying rational expressions.

Q&A

Q: What is the first step in simplifying a rational expression?

A: The first step in simplifying a rational expression is to invert and multiply the fractions. This means that we need to change the division sign to a multiplication sign and change the second fraction to its reciprocal.

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

A: To simplify a rational expression with multiple terms in the numerator and denominator, we need to follow the same steps as before. We need to invert and multiply the fractions, simplify the expression, check for any common factors, and cancel out common factors.

Q: What is the difference between adding and subtracting rational expressions and multiplying and dividing rational expressions?

A: The main difference between adding and subtracting rational expressions and multiplying and dividing rational expressions is the operation that we are performing. When we add or subtract rational expressions, we need to find a common denominator and then combine the numerators. When we multiply or divide rational expressions, we need to follow the rules for multiplying and dividing fractions.

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

A: To simplify a rational expression with a variable in the denominator, we need to follow the same steps as before. We need to invert and multiply the fractions, simplify the expression, check for any common factors, and cancel out common factors.

Q: What is the final step in simplifying a rational expression?

A: The final step in simplifying a rational expression is to check if there are any common factors in the numerator and denominator that can be canceled out. If there are, we need to cancel them out to simplify the expression.

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

A: Yes, we can simplify a rational expression with a negative exponent. To do this, we need to follow the same steps as before, but we need to be careful when simplifying the expression.

Q: How do I simplify a rational expression with a fraction in the numerator or denominator?

A: To simplify a rational expression with a fraction in the numerator or denominator, we need to follow the same steps as before. We need to invert and multiply the fractions, simplify the expression, check for any common factors, and cancel out common factors.

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

A: The main difference between a rational expression and a rational number is that a rational expression is a fraction that contains variables and constants in the numerator and denominator, while a rational number is a fraction that contains only integers in the numerator and denominator.

Conclusion

In conclusion, simplifying rational expressions can be a challenging task, but with the right approach, it can be simplified to a manageable form. By following the steps outlined in this article, you can simplify rational expressions with multiple terms in the numerator and denominator, variables in the denominator, and fractions in the numerator or denominator.

Final Tips

• Always follow the order of operations when simplifying rational expressions.
• Be careful when simplifying rational expressions with negative exponents.
• Check for any common factors in the numerator and denominator before simplifying the expression.
• Use a calculator to check your work and ensure that the expression is simplified correctly.

Final Answer

The final answer is $\boxed{\frac{2x^2 - 5x + 2}{x^2 + 6x + 8}}$.