Simplify The Expression:$\[ \frac{3x^2 + 3x}{8x^3 + 27} \div \frac{x+1}{2x^2 + X - 3} \times \frac{4x^2 - 6x + 9}{x-1} \\]

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will delve into the world of algebra and explore the process of simplifying a complex expression. We will break down the expression into manageable parts, apply various algebraic techniques, and ultimately arrive at a simplified form.

The Expression to Simplify

The given expression is:

$$\frac{3x^2 + 3x}{8x^3 + 27} \div \frac{x+1}{2x^2 + x - 3} \times \frac{4x^2 - 6x + 9}{x-1}$$

Step 1: Factorize the Numerators and Denominators

To simplify the expression, we need to factorize the numerators and denominators of each fraction. Let's start with the first fraction:

$$\frac{3x^2 + 3x}{8x^3 + 27}$$

We can factor out the greatest common factor (GCF) from the numerator and denominator:

$$\frac{3x(x + 1)}{8x^3 + 27}$$

Now, let's factorize the denominator:

$$8x^3 + 27 = (2x)^3 + 3^3$$

Using the sum of cubes formula, we can rewrite the denominator as:

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

Simplifying further, we get:

$$(2x + 3)(4x^2 - 6x + 9)$$

So, the first fraction becomes:

$$\frac{3x(x + 1)}{(2x + 3)(4x^2 - 6x + 9)}$$

Step 2: Factorize the Second Fraction

Now, let's move on to the second fraction:

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

We can factorize the numerator and denominator:

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

Step 3: Factorize the Third Fraction

The third fraction is:

$$\frac{4x^2 - 6x + 9}{x-1}$$

We can factorize the numerator:

$$4x^2 - 6x + 9 = (2x - 3)^2$$

So, the third fraction becomes:

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

Step 4: Simplify the Expression

Now that we have factorized all the fractions, we can simplify the expression by canceling out common factors:

$$\frac{3x(x + 1)}{(2x + 3)(4x^2 - 6x + 9)} \div \frac{x+1}{(x+3)(2x-1)} \times \frac{(2x - 3)^2}{x-1}$$

We can cancel out the common factor $(x+1)$ from the first and second fractions:

$$\frac{3x}{(2x + 3)(4x^2 - 6x + 9)} \div \frac{1}{(x+3)(2x-1)} \times \frac{(2x - 3)^2}{x-1}$$

Step 5: Simplify the Division

To simplify the division, we can multiply the first fraction by the reciprocal of the second fraction:

$$\frac{3x}{(2x + 3)(4x^2 - 6x + 9)} \times \frac{(x+3)(2x-1)}{1} \times \frac{(2x - 3)^2}{x-1}$$

Step 6: Simplify the Multiplication

Now, let's simplify the multiplication by canceling out common factors:

$$\frac{3x(x+3)(2x-1)(2x-3)^2}{(2x + 3)(4x^2 - 6x + 9)(x-1)}$$

Step 7: Simplify the Expression Further

We can simplify the expression further by canceling out common factors:

$$\frac{3x(x+3)(2x-1)(2x-3)^2}{(2x + 3)(4x^2 - 6x + 9)(x-1)} = \frac{3x(x+3)(2x-1)(2x-3)^2}{(2x + 3)(2x-3)^2(x-1)}$$

Canceling out the common factor $(2x-3)^2$, we get:

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

Step 8: Simplify the Expression Even Further

We can simplify the expression even further by canceling out common factors:

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

Canceling out the common factor $(x+1)$, we get:

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

Step 9: Simplify the Expression to Its Final Form

Finally, we can simplify the expression to its final form by canceling out common factors:

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

The final simplified expression is:

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

Conclusion

In this article, we have simplified a complex expression by applying various algebraic techniques, including factorization, cancellation, and multiplication. We have broken down the expression into manageable parts, identified common factors, and canceled them out to arrive at a simplified form. The final simplified expression is:

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

This expression can be further simplified by canceling out common factors, but for the purpose of this article, we have arrived at a simplified form that is easy to understand and work with.

Final Answer

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

Introduction

In our previous article, we simplified a complex expression by applying various algebraic techniques, including factorization, cancellation, and multiplication. In this article, we will provide a Q&A guide to help you understand the process of simplifying expressions and answer some common questions related to algebraic manipulation.

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to factorize the numerators and denominators of each fraction. This will help you identify common factors and cancel them out to simplify the expression.

Q: How do I factorize a numerator or denominator?

A: To factorize a numerator or denominator, you need to identify the greatest common factor (GCF) and factor it out. You can also use the sum of cubes formula to factorize expressions of the form $a^3 + b^3$.

Q: What is the difference between factorization and cancellation?

A: Factorization is the process of breaking down an expression into its prime factors, while cancellation is the process of eliminating common factors between two or more expressions.

Q: How do I simplify a division of fractions?

A: To simplify a division of fractions, you need to multiply the first fraction by the reciprocal of the second fraction. This will help you eliminate the division sign and simplify the expression.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{3x(x+3)(2x-1)}{(2x + 3)(x-1)}$.

Q: Can I simplify the expression further?

A: Yes, you can simplify the expression further by canceling out common factors. However, for the purpose of this article, we have arrived at a simplified form that is easy to understand and work with.

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

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

• Not factorizing the numerators and denominators
• Not canceling out common factors
• Not simplifying the expression further
• Not checking for common factors between two or more expressions

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

• Calculating probabilities and statistics
• Modeling population growth and decay
• Solving optimization problems
• Analyzing data and making predictions

Conclusion

In this article, we have provided a Q&A guide to help you understand the process of simplifying expressions and answer some common questions related to algebraic manipulation. We have also discussed some common mistakes to avoid and provided some tips for practicing and improving your skills. By following these tips and practicing regularly, you can become proficient in simplifying expressions and apply your skills to real-world problems.

Final Answer

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