Apply The Distributive Property To Factor Out The Greatest Common Factor Of All Three Terms.$\[9a - 18b + 21c = \square\\]

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Introduction

In mathematics, the distributive property is a fundamental concept that allows us to expand and simplify algebraic expressions. It states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac. In this article, we will apply the distributive property to factor out the greatest common factor (GCF) of all three terms in the given equation.

Understanding the Distributive Property

The distributive property is a powerful tool that enables us to simplify complex algebraic expressions. It allows us to expand and combine like terms, making it easier to solve equations and manipulate expressions. The distributive property can be applied in two ways:

  • Distributing a single term: This involves multiplying a single term by each term inside the parentheses.
  • Distributing multiple terms: This involves multiplying multiple terms by each term inside the parentheses.

Applying the Distributive Property to Factor Out the GCF

To factor out the GCF of all three terms, we need to identify the common factors among the terms. In this case, the GCF of 9a, 18b, and 21c is 3. We can apply the distributive property to factor out the GCF as follows:

9a−18b+21c=3(3a−6b+7c){9a - 18b + 21c = 3(3a - 6b + 7c)}

Step-by-Step Solution

To factor out the GCF, we can follow these steps:

  1. Identify the GCF: The GCF of 9a, 18b, and 21c is 3.
  2. Distribute the GCF: We can distribute the GCF to each term inside the parentheses.
  3. Simplify the expression: After distributing the GCF, we can simplify the expression by combining like terms.

Example

Let's consider an example to illustrate the concept. Suppose we have the equation:

12x−24y+36z=□{12x - 24y + 36z = \square}

To factor out the GCF, we can apply the distributive property as follows:

12x−24y+36z=12(x−2y+3z){12x - 24y + 36z = 12(x - 2y + 3z)}

Conclusion

In this article, we applied the distributive property to factor out the greatest common factor of all three terms in the given equation. We identified the GCF, distributed it to each term inside the parentheses, and simplified the expression. The distributive property is a powerful tool that enables us to simplify complex algebraic expressions and solve equations. By applying the distributive property, we can factor out the GCF and make it easier to manipulate expressions.

Common Mistakes to Avoid

When applying the distributive property, there are several common mistakes to avoid:

  • Not identifying the GCF: Failing to identify the GCF can lead to incorrect factorization.
  • Not distributing the GCF: Failing to distribute the GCF to each term inside the parentheses can lead to incorrect factorization.
  • Not simplifying the expression: Failing to simplify the expression after distributing the GCF can lead to incorrect factorization.

Tips and Tricks

Here are some tips and tricks to help you apply the distributive property:

  • Use the distributive property to simplify complex expressions: The distributive property can help you simplify complex expressions and make them easier to manipulate.
  • Use the distributive property to factor out the GCF: The distributive property can help you factor out the GCF and make it easier to manipulate expressions.
  • Use the distributive property to solve equations: The distributive property can help you solve equations by simplifying complex expressions and making them easier to manipulate.

Practice Problems

Here are some practice problems to help you apply the distributive property:

  • Factor out the GCF of 12x, 24y, and 36z: Use the distributive property to factor out the GCF of 12x, 24y, and 36z.
  • Factor out the GCF of 9a, 18b, and 21c: Use the distributive property to factor out the GCF of 9a, 18b, and 21c.
  • Solve the equation 12x - 24y + 36z = 0: Use the distributive property to solve the equation 12x - 24y + 36z = 0.

Conclusion

Q: What is the distributive property?

A: The distributive property is a fundamental concept in mathematics that allows us to expand and simplify algebraic expressions. It states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to identify the common factors among the terms and distribute them to each term inside the parentheses. You can then simplify the expression by combining like terms.

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

A: The GCF is the largest number that divides all the terms in an expression without leaving a remainder. In the given equation, the GCF of 9a, 18b, and 21c is 3.

Q: How do I factor out the GCF?

A: To factor out the GCF, you need to distribute the GCF to each term inside the parentheses. You can then simplify the expression by combining like terms.

Q: What are some common mistakes to avoid when applying the distributive property?

A: Some common mistakes to avoid when applying the distributive property include:

  • Not identifying the GCF
  • Not distributing the GCF to each term inside the parentheses
  • Not simplifying the expression after distributing the GCF

Q: How do I simplify complex expressions using the distributive property?

A: To simplify complex expressions using the distributive property, you need to identify the common factors among the terms and distribute them to each term inside the parentheses. You can then simplify the expression by combining like terms.

Q: Can I use the distributive property to solve equations?

A: Yes, you can use the distributive property to solve equations. By simplifying complex expressions and making them easier to manipulate, you can solve equations more efficiently.

Q: What are some tips and tricks for applying the distributive property?

A: Some tips and tricks for applying the distributive property include:

  • Using the distributive property to simplify complex expressions
  • Using the distributive property to factor out the GCF
  • Using the distributive property to solve equations

Q: How do I practice applying the distributive property?

A: You can practice applying the distributive property by working through practice problems, such as factoring out the GCF of 12x, 24y, and 36z, or solving the equation 12x - 24y + 36z = 0.

Q: What are some real-world applications of the distributive property?

A: The distributive property has many real-world applications, including:

  • Simplifying complex expressions in algebra and calculus
  • Factoring out the GCF in polynomial expressions
  • Solving equations in physics and engineering

Conclusion

In this article, we answered some frequently asked questions about the distributive property, including how to apply it, what the greatest common factor is, and how to simplify complex expressions. We also provided some tips and tricks for applying the distributive property and discussed some real-world applications of the concept. By understanding the distributive property and how to apply it, you can simplify complex expressions and solve equations more efficiently.