Simplify: X 2 − X − 6 X 2 − 4 X + 3 ÷ X 2 − 2 X − 8 X 2 + 3 X − 4 = \frac{x^2-x-6}{x^2-4x+3} \div \frac{x^2-2x-8}{x^2+3x-4} = X 2 − 4 X + 3 X 2 − X − 6 ​ ÷ X 2 + 3 X − 4 X 2 − 2 X − 8 ​ =

$$

Introduction

In this article, we will delve into the world of algebraic expressions and simplify a complex division problem involving two rational expressions. Rational expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. The given problem involves dividing two rational expressions, and we will use various techniques to simplify 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 factoring the numerator and denominator, canceling out common factors, and then simplifying the resulting expression. In this problem, we have two rational expressions:

$\frac{x^2-x-6}{x^2-4x+3}$ and $\frac{x^2-2x-8}{x^2+3x-4}$

Step 1: Factor the Numerator and Denominator

To simplify the given problem, we need to factor the numerator and denominator of both rational expressions.

Factor the Numerator and Denominator of the First Rational Expression

The numerator of the first rational expression is $x^2-x-6$, which can be factored as $(x-3)(x+2)$. The denominator is $x^2-4x+3$, which can be factored as $(x-3)(x-1)$.

import sympy as sp
x = sp.symbols('x')

numerator_1 = x2 - x - 6
denominator_1 = x2 - 4*x + 3

factored_numerator_1 = sp.factor(numerator_1)
factored_denominator_1 = sp.factor(denominator_1)
print(factored_numerator_1)
print(factored_denominator_1)

Factor the Numerator and Denominator of the Second Rational Expression

The numerator of the second rational expression is $x^2-2x-8$, which can be factored as $(x-4)(x+2)$. The denominator is $x^2+3x-4$, which can be factored as $(x+4)(x-1)$.

# Define the numerator and denominator of the second rational expression
numerator_2 = x**2 - 2*x - 8
denominator_2 = x**2 + 3*x - 4

factored_numerator_2 = sp.factor(numerator_2)
factored_denominator_2 = sp.factor(denominator_2)
print(factored_numerator_2)
print(factored_denominator_2)

Step 2: Cancel Out Common Factors

Now that we have factored the numerator and denominator of both rational expressions, we can cancel out common factors.

Cancel Out Common Factors of the First Rational Expression

The first rational expression is $\frac{(x-3)(x+2)}{(x-3)(x-1)}$. We can cancel out the common factor $(x-3)$ from the numerator and denominator.

# Cancel out the common factor (x-3)
simplified_expression_1 = (x+2)/(x-1)
print(simplified_expression_1)

Cancel Out Common Factors of the Second Rational Expression

The second rational expression is $\frac{(x-4)(x+2)}{(x+4)(x-1)}$. We can cancel out the common factor $(x+2)$ from the numerator and denominator.

# Cancel out the common factor (x+2)
simplified_expression_2 = (x-4)/(x+4)
print(simplified_expression_2)

Step 3: Simplify the Resulting Expression

Now that we have canceled out common factors, we can simplify the resulting expression.

Simplify the Resulting Expression

The resulting expression is $\frac{(x+2)}{(x-1)} \div \frac{(x-4)}{(x+4)}$. We can simplify this expression by multiplying the numerator and denominator of the first rational expression by the reciprocal of the second rational expression.

# Simplify the resulting expression
simplified_expression = ((x+2)/(x-1)) * ((x+4)/(x-4))
print(simplified_expression)

Conclusion

In this article, we simplified a complex division problem involving two rational expressions. We used various techniques such as factoring the numerator and denominator, canceling out common factors, and simplifying the resulting expression. The final simplified expression is $\frac{(x+2)(x+4)}{(x-1)(x-4)}$. This expression can be further simplified by factoring the numerator and denominator.

Final Answer

$$ $$

Introduction

In our previous article, we simplified a complex division problem involving two rational expressions. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

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

A: The first step in simplifying a rational expression is to factor the numerator and denominator.

Q: How do I factor a rational expression?

A: To factor a rational expression, you need to find the greatest common factor (GCF) of the numerator and denominator. You can then factor the GCF out of both the numerator and denominator.

Q: What is the difference between factoring and canceling out common factors?

A: Factoring involves breaking down a rational expression into its simplest form by finding the GCF and factoring it out. Canceling out common factors involves removing any common factors that appear in both the numerator and denominator.

Q: How do I cancel out common factors in a rational expression?

A: To cancel out common factors in a rational expression, you need to identify any common factors that appear in both the numerator and denominator. You can then remove these common factors to simplify the expression.

Q: What is the final simplified expression for the given problem?

A: The final simplified expression for the given problem is $\frac{(x+2)(x+4)}{(x-1)(x-4)}$.

Q: Can I further simplify the final expression?

A: Yes, you can further simplify the final expression by factoring the numerator and denominator.

Q: How do I factor the numerator and denominator of a rational expression?

A: To factor the numerator and denominator of a rational expression, you need to find the GCF of the numerator and denominator. You can then factor the GCF out of both the numerator and denominator.

Q: What is the importance of simplifying rational expressions?

A: Simplifying rational expressions is important because it helps to make the expression easier to work with and understand. It also helps to eliminate any unnecessary complexity in the expression.

Q: Can I use technology to simplify rational expressions?

A: Yes, you can use technology such as calculators or computer software to simplify rational expressions. However, it's always a good idea to understand the underlying math concepts and be able to simplify expressions by hand.

Conclusion

In this Q&A article, we provided answers to common questions related to simplifying rational expressions. We hope that this article has helped to clarify any doubts and provide additional information on the topic.

Final Answer

The final answer is $\boxed{\frac{(x+2)(x+4)}{(x-1)(x-4)}}$.