Multiply The Following Expression And Simplify Your Answer As Much As Possible: X 2 − 9 X − 1 ⋅ 4 X − 4 X 2 − X − 6 \frac{x^2-9}{x-1} \cdot \frac{4x-4}{x^2-x-6} X − 1 X 2 − 9 ​ ⋅ X 2 − X − 6 4 X − 4 ​ Simplified Answer: □ \square □

Introduction

In this article, we will be multiplying the given expression and simplifying the answer as much as possible. The expression given is $\frac{x^2-9}{x-1} \cdot \frac{4x-4}{x^2-x-6}$. We will use the rules of algebra and simplification to simplify the expression and find the final answer.

Step 1: Factor the Numerators and Denominators

To simplify the expression, we need to factor the numerators and denominators of both fractions. The numerator of the first fraction can be factored as $x^2-9 = (x-3)(x+3)$, and the denominator can be factored as $x-1$. The numerator of the second fraction can be factored as $4x-4 = 4(x-1)$, and the denominator can be factored as $x^2-x-6 = (x-3)(x+2)$.

Step 2: Cancel Out Common Factors

Now that we have factored the numerators and denominators, we can cancel out common factors between the two fractions. The common factor between the two fractions is $(x-3)$, which can be canceled out from both the numerator and denominator.

Step 3: Simplify the Expression

After canceling out the common factor, the expression becomes $\frac{(x+3) \cdot 4(x-1)}{(x-1)(x+2)}$. We can further simplify the expression by canceling out the common factor $(x-1)$ from the numerator and denominator.

Step 4: Final Simplification

After canceling out the common factor $(x-1)$, the expression becomes $\frac{4(x+3)}{(x+2)}$. This is the final simplified expression.

Conclusion

In this article, we multiplied the given expression and simplified the answer as much as possible. We used the rules of algebra and simplification to factor the numerators and denominators, cancel out common factors, and simplify the expression. The final simplified expression is $\frac{4(x+3)}{(x+2)}$.

Final Answer

The final answer is $\boxed{\frac{4(x+3)}{(x+2)}}$.

Discussion

The given expression is a product of two rational expressions. To simplify the expression, we need to factor the numerators and denominators of both fractions and cancel out common factors. The final simplified expression is $\frac{4(x+3)}{(x+2)}$. This expression can be further simplified by canceling out the common factor $(x+2)$ from the numerator and denominator, but only if $x \neq -2$.

Example

Let's consider an example to illustrate the concept. Suppose we have the expression $\frac{x^2-9}{x-1} \cdot \frac{4x-4}{x^2-x-6}$, and we want to simplify it. We can follow the steps outlined above to simplify the expression.

Step 1: Factor the Numerators and Denominators

The numerator of the first fraction can be factored as $x^2-9 = (x-3)(x+3)$, and the denominator can be factored as $x-1$. The numerator of the second fraction can be factored as $4x-4 = 4(x-1)$, and the denominator can be factored as $x^2-x-6 = (x-3)(x+2)$.

Step 2: Cancel Out Common Factors

The common factor between the two fractions is $(x-3)$, which can be canceled out from both the numerator and denominator.

Step 3: Simplify the Expression

After canceling out the common factor, the expression becomes $\frac{(x+3) \cdot 4(x-1)}{(x-1)(x+2)}$. We can further simplify the expression by canceling out the common factor $(x-1)$ from the numerator and denominator.

Step 4: Final Simplification

After canceling out the common factor $(x-1)$, the expression becomes $\frac{4(x+3)}{(x+2)}$. This is the final simplified expression.

Conclusion

In this example, we multiplied the given expression and simplified the answer as much as possible. We used the rules of algebra and simplification to factor the numerators and denominators, cancel out common factors, and simplify the expression. The final simplified expression is $\frac{4(x+3)}{(x+2)}$.

Final Answer

The final answer is $\boxed{\frac{4(x+3)}{(x+2)}}$.

Applications

The concept of multiplying and simplifying rational expressions has numerous applications in mathematics and other fields. Some of the applications include:

• Algebra: Rational expressions are used extensively in algebra to solve equations and inequalities.
• Calculus: Rational expressions are used to find derivatives and integrals of functions.
• Engineering: Rational expressions are used to model and analyze complex systems in engineering.
• Economics: Rational expressions are used to model and analyze economic systems.

Conclusion

In conclusion, multiplying and simplifying rational expressions is an essential skill in mathematics and other fields. By following the steps outlined above, we can simplify complex expressions and find the final answer. The final simplified expression is $\frac{4(x+3)}{(x+2)}$. This expression can be further simplified by canceling out the common factor $(x+2)$ from the numerator and denominator, but only if $x \neq -2$.

Final Answer

The final answer is $\boxed{\frac{4(x+3)}{(x+2)}}$.

Introduction

In this article, we will be answering some frequently asked questions about multiplying and simplifying rational expressions. The expression given is $\frac{x^2-9}{x-1} \cdot \frac{4x-4}{x^2-x-6}$. We will use the rules of algebra and simplification to simplify the expression and find the final answer.

Q: What is the first step in multiplying and simplifying rational expressions?

A: The first step in multiplying and simplifying rational expressions is to factor the numerators and denominators of both fractions.

Q: How do I factor the numerators and denominators?

A: To factor the numerators and denominators, we need to find the greatest common factor (GCF) of the terms and then factor out the GCF.

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

A: The greatest common factor (GCF) is the largest factor that divides all the terms in a set of numbers.

Q: How do I cancel out common factors?

A: To cancel out common factors, we need to identify the common factors between the two fractions and then cancel them out.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{4(x+3)}{(x+2)}$.

Q: Can I further simplify the expression?

A: Yes, you can further simplify the expression by canceling out the common factor $(x+2)$ from the numerator and denominator, but only if $x \neq -2$.

Q: What are some applications of multiplying and simplifying rational expressions?

A: Some applications of multiplying and simplifying rational expressions include algebra, calculus, engineering, and economics.

Q: Why is it important to simplify rational expressions?

A: Simplifying rational expressions is important because it helps us to find the final answer and understand the underlying mathematics.

Q: How do I know when to cancel out common factors?

A: You know when to cancel out common factors when you have identified the common factors between the two fractions.

Q: What is the final answer?

A: The final answer is $\boxed{\frac{4(x+3)}{(x+2)}}$.

Q: Can I use this method to simplify other rational expressions?

A: Yes, you can use this method to simplify other rational expressions by following the same steps.

Q: What if I have a rational expression with multiple variables?

A: If you have a rational expression with multiple variables, you can still use the same method to simplify it.

Q: How do I know if I have simplified the expression correctly?

A: You can check if you have simplified the expression correctly by plugging in some values for the variables and checking if the expression simplifies to the expected value.

Q: What if I get stuck while simplifying a rational expression?

A: If you get stuck while simplifying a rational expression, you can try breaking it down into smaller steps or seeking help from a teacher or tutor.

Conclusion

In conclusion, multiplying and simplifying rational expressions is an essential skill in mathematics and other fields. By following the steps outlined above, we can simplify complex expressions and find the final answer. The final simplified expression is $\frac{4(x+3)}{(x+2)}$. This expression can be further simplified by canceling out the common factor $(x+2)$ from the numerator and denominator, but only if $x \neq -2$.

Final Answer

The final answer is $\boxed{\frac{4(x+3)}{(x+2)}}$.