Simplify The Expression:$\[ \frac{9x^2(x^3-1)-4(x^3-1)}{3x^2-x-2} \div \frac{3x^2+3x+3}{9x+6} \\]

Introduction

Algebraic expressions can be complex and daunting, but with the right techniques and strategies, they can be simplified to reveal their underlying structure. In this article, we will delve into the world of algebraic manipulation and explore the steps involved in simplifying a given expression. We will use the expression $\frac{9x^2(x^3-1)-4(x^3-1)}{3x^2-x-2} \div \frac{3x^2+3x+3}{9x+6}$ as a case study and demonstrate how to simplify it using various algebraic techniques.

Understanding the Expression

Before we begin simplifying the expression, it's essential to understand its structure and components. The given expression is a rational expression, which means it is the ratio of two polynomials. The numerator is $9x^2(x^3-1)-4(x^3-1)$, and the denominator is $3x^2-x-2$. The expression is also divided by another rational expression, $\frac{3x^2+3x+3}{9x+6}$.

Factoring the Numerator

To simplify the expression, we can start by factoring the numerator. Factoring involves expressing a polynomial as a product of simpler polynomials. In this case, we can factor out the common term $(x^3-1)$ from the numerator:

$$9x^2(x^3-1)-4(x^3-1) = (x^3-1)(9x^2-4)$$

This simplifies the numerator and makes it easier to work with.

Factoring the Denominator

Next, we can factor the denominator. The denominator is a quadratic expression, and we can factor it using the quadratic formula or by finding two numbers that multiply to $-6$ and add to $-1$. In this case, we can factor the denominator as follows:

$$3x^2-x-2 = (3x+2)(x-1)$$

This simplifies the denominator and makes it easier to work with.

Simplifying the Expression

Now that we have factored the numerator and denominator, we can simplify the expression. We can start by canceling out any common factors between the numerator and denominator. In this case, we can cancel out the factor $(x^3-1)$:

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

Next, we can simplify the expression by canceling out any common factors between the numerator and denominator of the second rational expression. In this case, we can cancel out the factor $3$:

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

Canceling Out Common Factors

Now that we have simplified the expression, we can cancel out any common factors between the numerator and denominator. In this case, we can cancel out the factor $(x+1)$:

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

Simplifying the Expression Further

Now that we have canceled out any common factors, we can simplify the expression further. We can start by multiplying the numerators and denominators:

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

Final Simplification

Finally, we can simplify the expression by combining like terms in the numerator and denominator. In this case, we can combine the terms as follows:

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

Conclusion

In this article, we have demonstrated how to simplify a given algebraic expression using various techniques and strategies. We have factored the numerator and denominator, canceled out common factors, and simplified the expression further by combining like terms. The final simplified expression is $\frac{3(x^3-1)(9x^2-4)(x+1)}{(3x+2)(x-1)(x^2+x+1)}$. This expression is a simplified version of the original expression and reveals its underlying structure.

Tips and Tricks

• When simplifying algebraic expressions, it's essential to factor the numerator and denominator to reveal any common factors.
• Canceling out common factors can simplify the expression and make it easier to work with.
• Combining like terms in the numerator and denominator can further simplify the expression.
• When working with rational expressions, it's essential to consider the signs of the numerator and denominator.

Common Mistakes to Avoid

• Failing to factor the numerator and denominator can lead to a more complex expression.
• Not canceling out common factors can result in a more complicated expression.
• Not combining like terms in the numerator and denominator can lead to a more complex expression.

Final Thoughts

Simplifying algebraic expressions is an essential skill in mathematics, and it requires patience, practice, and persistence. By following the techniques and strategies outlined in this article, you can simplify even the most complex algebraic expressions and reveal their underlying structure. Remember to factor the numerator and denominator, cancel out common factors, and combine like terms to simplify the expression. With practice and experience, you will become proficient in simplifying algebraic expressions and be able to tackle even the most challenging problems.

Introduction

In our previous article, we demonstrated how to simplify a given algebraic expression using various techniques and strategies. We factored the numerator and denominator, canceled out common factors, and simplified the expression further by combining like terms. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

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

A: The first step in simplifying an algebraic expression is to factor the numerator and denominator. Factoring involves expressing a polynomial as a product of simpler polynomials. By factoring the numerator and denominator, you can reveal any common factors that can be canceled out.

Q: How do I factor a polynomial?

A: Factoring a polynomial involves finding two or more polynomials that multiply together to give the original polynomial. You can use various techniques such as grouping, synthetic division, or the quadratic formula to factor a polynomial.

Q: What is the difference between factoring and simplifying an algebraic expression?

A: Factoring involves expressing a polynomial as a product of simpler polynomials, while simplifying an algebraic expression involves canceling out common factors and combining like terms to reveal the underlying structure of the expression.

Q: Can I simplify an algebraic expression by canceling out common factors without factoring the numerator and denominator?

A: No, you cannot simplify an algebraic expression by canceling out common factors without factoring the numerator and denominator. Factoring the numerator and denominator is essential to reveal any common factors that can be canceled out.

Q: How do I know if I have canceled out all the common factors in an algebraic expression?

A: To ensure that you have canceled out all the common factors in an algebraic expression, you should check the numerator and denominator carefully. If you have canceled out all the common factors, the numerator and denominator should be relatively prime, meaning they have no common factors other than 1.

Q: Can I simplify an algebraic expression by combining like terms without factoring the numerator and denominator?

A: No, you cannot simplify an algebraic expression by combining like terms without factoring the numerator and denominator. Combining like terms is an essential step in simplifying an algebraic expression, but it should be done after factoring the numerator and denominator.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to check the expression for any remaining common factors. If you have canceled out all the common factors, the expression is simplified.

Q: Can I simplify an algebraic expression using a calculator?

A: Yes, you can simplify an algebraic expression using a calculator. However, it's essential to understand the underlying mathematics and techniques involved in simplifying an algebraic expression. Using a calculator can help you verify your work, but it's not a substitute for understanding the mathematics.

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

A: To ensure that you have simplified an algebraic expression correctly, you should check the expression carefully. You should verify that you have canceled out all the common factors and combined like terms correctly. If you are unsure, you can use a calculator to verify your work.

Q: Can I simplify an algebraic expression that has multiple variables?

A: Yes, you can simplify an algebraic expression that has multiple variables. However, it's essential to understand the underlying mathematics and techniques involved in simplifying an algebraic expression with multiple variables.

Q: How do I simplify an algebraic expression with multiple variables?

A: To simplify an algebraic expression with multiple variables, you should follow the same steps as simplifying an algebraic expression with a single variable. You should factor the numerator and denominator, cancel out common factors, and combine like terms.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it requires patience, practice, and persistence. By following the techniques and strategies outlined in this article, you can simplify even the most complex algebraic expressions and reveal their underlying structure. Remember to factor the numerator and denominator, cancel out common factors, and combine like terms to simplify the expression. With practice and experience, you will become proficient in simplifying algebraic expressions and be able to tackle even the most challenging problems.

Tips and Tricks

• When simplifying algebraic expressions, it's essential to factor the numerator and denominator to reveal any common factors.
• Canceling out common factors can simplify the expression and make it easier to work with.
• Combining like terms in the numerator and denominator can further simplify the expression.
• When working with rational expressions, it's essential to consider the signs of the numerator and denominator.

Common Mistakes to Avoid

• Failing to factor the numerator and denominator can lead to a more complex expression.
• Not canceling out common factors can result in a more complicated expression.
• Not combining like terms in the numerator and denominator can lead to a more complex expression.

Final Thoughts

Simplifying algebraic expressions is an essential skill in mathematics, and it requires patience, practice, and persistence. By following the techniques and strategies outlined in this article, you can simplify even the most complex algebraic expressions and reveal their underlying structure. Remember to factor the numerator and denominator, cancel out common factors, and combine like terms to simplify the expression. With practice and experience, you will become proficient in simplifying algebraic expressions and be able to tackle even the most challenging problems.