Factor Completely 4 A B + 6 A C − 6 B D − 9 C D 4ab + 6ac - 6bd - 9cd 4 Ab + 6 A C − 6 B D − 9 C D .

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Introduction

Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. In this article, we will focus on factoring the expression $4ab + 6ac - 6bd - 9cd$. Factoring this expression will help us simplify it and make it easier to work with.

Understanding the Expression

Before we start factoring, let's take a closer look at the expression $4ab + 6ac - 6bd - 9cd$. This expression consists of four terms, each with a combination of variables and constants. To factor this expression, we need to identify the common factors among the terms.

Identifying Common Factors

The first step in factoring is to identify the common factors among the terms. In this case, we can see that the terms $4ab$ and $6ac$ have a common factor of $2a$, while the terms $-6bd$ and $-9cd$ have a common factor of $-3d$. We can also see that the terms $4ab$ and $-6bd$ have a common factor of $2b$, while the terms $6ac$ and $-9cd$ have a common factor of $3c$.

Factoring by Grouping

Now that we have identified the common factors, we can start factoring by grouping. We can group the terms $4ab$ and $6ac$ together, and the terms $-6bd$ and $-9cd$ together. This gives us:

$$4ab + 6ac - 6bd - 9cd = (4ab + 6ac) - (6bd + 9cd)$$

Factoring Out Common Factors

Now that we have grouped the terms, we can factor out the common factors. We can factor out $2a$ from the first group, and $-3d$ from the second group. This gives us:

$$4ab + 6ac - 6bd - 9cd = 2a(2b + 3c) - 3d(2b + 3c)$$

Factoring Out the Common Binomial Factor

Now that we have factored out the common factors, we can factor out the common binomial factor $(2b + 3c)$. This gives us:

$$4ab + 6ac - 6bd - 9cd = (2b + 3c)(2a - 3d)$$

Conclusion

In this article, we have factored the expression $4ab + 6ac - 6bd - 9cd$ completely. We started by identifying the common factors among the terms, and then factored by grouping. Finally, we factored out the common binomial factor to get the final factored form of the expression.

Final Answer

The final answer is: $\boxed{(2b + 3c)(2a - 3d)}$

Tips and Tricks

• When factoring, always look for common factors among the terms.
• Use grouping to factor out common factors.
• Factor out the common binomial factor to get the final factored form of the expression.

Common Mistakes to Avoid

• Don't forget to factor out the common binomial factor.
• Make sure to group the terms correctly before factoring.
• Don't factor out a common factor that is not present in all the terms.

Real-World Applications

Factoring is a fundamental concept in algebra that has many real-world applications. For example, factoring can be used to simplify complex expressions in physics and engineering, and to solve systems of equations in economics and finance.

Conclusion

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Introduction

In our previous article, we factored the expression $4ab + 6ac - 6bd - 9cd$ completely. However, we know that understanding the concept of factoring can be a challenging task for many students. In this article, we will answer some of the most frequently asked questions about factoring the expression $4ab + 6ac - 6bd - 9cd$.

Q: What is factoring?

A: Factoring is a process of expressing an algebraic expression as a product of simpler expressions. In other words, it involves breaking down a complex expression into smaller, more manageable parts.

Q: Why is factoring important?

A: Factoring is an important concept in algebra because it allows us to simplify complex expressions and solve systems of equations. By factoring, we can identify the underlying structure of an expression and make it easier to work with.

Q: How do I factor the expression $4ab + 6ac - 6bd - 9cd$?

A: To factor the expression $4ab + 6ac - 6bd - 9cd$, we need to identify the common factors among the terms. We can group the terms $4ab$ and $6ac$ together, and the terms $-6bd$ and $-9cd$ together. Then, we can factor out the common factors $2a$ and $-3d$ from each group. Finally, we can factor out the common binomial factor $(2b + 3c)$ to get the final factored form of the expression.

Q: What are some common mistakes to avoid when factoring?

A: Some common mistakes to avoid when factoring include:

• Not identifying the common factors among the terms
• Not grouping the terms correctly
• Factoring out a common factor that is not present in all the terms
• Not factoring out the common binomial factor

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

A: To check if you have factored the expression correctly, you can multiply the factored form of the expression back together to get the original expression. If the result is the same as the original expression, then you have factored it correctly.

Q: What are some real-world applications of factoring?

A: Factoring has many real-world applications, including:

• Simplifying complex expressions in physics and engineering
• Solving systems of equations in economics and finance
• Identifying the underlying structure of an expression in computer science and programming

Q: Can I factor expressions with variables and constants?

A: Yes, you can factor expressions with variables and constants. In fact, factoring is often used to simplify expressions with variables and constants.

Q: Can I factor expressions with negative coefficients?

A: Yes, you can factor expressions with negative coefficients. When factoring expressions with negative coefficients, be sure to include the negative sign in the factored form of the expression.

Conclusion

In conclusion, factoring is a powerful tool in algebra that can be used to simplify complex expressions and solve systems of equations. By understanding the concept of factoring and following the steps outlined in this article, you can factor the expression $4ab + 6ac - 6bd - 9cd$ correctly and apply it to real-world problems.

Final Answer

The final answer is: $\boxed{(2b + 3c)(2a - 3d)}$