Apply The Distributive Property To Factor Out The Greatest Common Factor.$27 - 18r = \square$

Introduction

In algebra, the distributive property is a fundamental concept that allows us to expand and simplify expressions. One of the key applications of the distributive property is to factor out the greatest common factor (GCF) from an expression. In this article, we will explore how to apply the distributive property to factor out the GCF from an expression, using the example of the equation $27 - 18r = \square$.

Understanding the Distributive Property

The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

This property allows us to expand and simplify expressions by distributing the multiplication operation to each term inside the parentheses.

Applying the Distributive Property to Factor Out the GCF

To factor out the GCF from an expression, we need to identify the common factors among the terms. In the equation $27 - 18r = \square$, we can see that both terms have a common factor of $9$. To factor out the GCF, we can use the distributive property to rewrite the expression as follows:

$$27 - 18r = 9(3 - 2r)$$

In this rewritten expression, we have factored out the GCF of $9$ from both terms. The remaining expression inside the parentheses is a simplified form of the original expression.

Step-by-Step Guide to Factoring Out the GCF

To factor out the GCF from an expression, follow these steps:

1. Identify the common factors: Look for the common factors among the terms in the expression.
2. Use the distributive property: Rewrite the expression using the distributive property, with the common factor as the factor to be distributed.
3. Simplify the expression: Simplify the expression by combining like terms and removing any unnecessary parentheses.

Example 2: Factoring Out the GCF from a More Complex Expression

Let's consider a more complex expression, such as $24x^2 + 36x + 12 = \square$. To factor out the GCF, we need to identify the common factors among the terms. In this case, the common factors are $12$ and $x$. We can use the distributive property to rewrite the expression as follows:

$$24x^2 + 36x + 12 = 12(2x^2 + 3x + 1)$$

In this rewritten expression, we have factored out the GCF of $12$ from all three terms.

Conclusion

In conclusion, applying the distributive property to factor out the greatest common factor is a powerful technique in algebra. By identifying the common factors among the terms and using the distributive property, we can simplify expressions and make them easier to work with. Whether you're working with simple or complex expressions, factoring out the GCF is an essential skill to master.

Common Mistakes to Avoid

When factoring out the GCF, there are several common mistakes to avoid:

• Not identifying the common factors: Make sure to carefully examine the terms in the expression to identify the common factors.
• Not using the distributive property: Use the distributive property to rewrite the expression, with the common factor as the factor to be distributed.
• Not simplifying the expression: Simplify the expression by combining like terms and removing any unnecessary parentheses.

Practice Problems

To practice factoring out the GCF, try the following problems:

1. Factor out the GCF from the expression $18x^2 + 24x + 12 = \square$.
2. Factor out the GCF from the expression $30x^2 + 40x + 10 = \square$.
3. Factor out the GCF from the expression $24x^2 + 36x + 12 = \square$.

Answer Key

1. $6(3x^2 + 4x + 2)$
2. $10(3x^2 + 4x + 1)$
3. $12(2x^2 + 3x + 1)$

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

A: The greatest common factor (GCF) is the largest positive integer that divides each of the numbers in a set of numbers without leaving a remainder.

Q: Why is it important to factor out the GCF?

A: Factoring out the GCF is an essential skill in algebra because it allows us to simplify expressions and make them easier to work with. By factoring out the GCF, we can identify the common factors among the terms and rewrite the expression in a more manageable form.

Q: How do I identify the common factors in an expression?

A: To identify the common factors in an expression, look for the terms that have the same factor. For example, in the expression $24x^2 + 36x + 12$, the common factors are $12$ and $x$.

Q: What is the distributive property, and how is it used to factor out the GCF?

A: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

To factor out the GCF, we use the distributive property to rewrite the expression, with the common factor as the factor to be distributed.

Q: Can I factor out the GCF from a complex expression?

A: Yes, you can factor out the GCF from a complex expression. To do this, identify the common factors among the terms and use the distributive property to rewrite the expression.

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

A: Some common mistakes to avoid when factoring out the GCF include:

• Not identifying the common factors
• Not using the distributive property
• Not simplifying the expression

Q: How do I simplify an expression after factoring out the GCF?

A: To simplify an expression after factoring out the GCF, combine like terms and remove any unnecessary parentheses.

Q: Can I use the GCF to factor out more than one term?

A: Yes, you can use the GCF to factor out more than one term. To do this, identify the common factors among the terms and use the distributive property to rewrite the expression.

Q: What is the difference between factoring out the GCF and factoring out a common binomial?

A: Factoring out the GCF involves identifying the common factors among the terms and using the distributive property to rewrite the expression. Factoring out a common binomial involves identifying a common binomial factor and using the distributive property to rewrite the expression.

Q: Can I use the GCF to factor out a negative number?

A: Yes, you can use the GCF to factor out a negative number. To do this, identify the common factors among the terms and use the distributive property to rewrite the expression.

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

A: Factoring out the GCF has many real-world applications, including:

• Simplifying algebraic expressions in science and engineering
• Factoring out common factors in finance and economics
• Identifying common factors in data analysis and statistics

Conclusion

In conclusion, factoring out the greatest common factor is an essential skill in algebra that allows us to simplify expressions and make them easier to work with. By identifying the common factors among the terms and using the distributive property, we can rewrite the expression in a more manageable form. Remember to avoid common mistakes and simplify the expression after factoring out the GCF.