Apply The Distributive Property To Factor Out The Greatest Common Factor.$\[ 60 - 40y = \square \\]

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Introduction

In mathematics, the distributive property is a fundamental concept that allows us to expand and simplify algebraic 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 a given expression.

Understanding the Distributive Property

The distributive property states that for any real numbers a, b, and c:

a(b + c) = ab + ac

This property allows us to distribute the multiplication operation over the addition operation. In other words, we can multiply each term inside the parentheses by the factor outside the parentheses.

Factoring Out the Greatest Common Factor

The greatest common factor (GCF) of a set of numbers is the largest number that divides each of the numbers without leaving a remainder. When we have an expression with multiple terms, we can use the distributive property to factor out the GCF from each term.

Let's consider the expression:

60 - 40y

We can see that both terms have a common factor of 20. To factor out the GCF, we can use the distributive property as follows:

60 - 40y = 20(3 - 2y)

In this example, we have factored out the GCF (20) from each term, leaving us with a simplified expression.

Step-by-Step Guide to Factoring Out the GCF

Here's a step-by-step guide to factoring out the GCF using the distributive property:

  1. Identify the GCF: The first step is to identify the greatest common factor of the terms in the expression.
  2. Distribute the GCF: Once we have identified the GCF, we can use the distributive property to distribute it over each term in the expression.
  3. Simplify the expression: After distributing the GCF, we can simplify the expression by combining like terms.

Example 1: Factoring Out the GCF from a Simple Expression

Let's consider the expression:

12x + 18y

We can see that both terms have a common factor of 6. To factor out the GCF, we can use the distributive property as follows:

12x + 18y = 6(2x + 3y)

In this example, we have factored out the GCF (6) from each term, leaving us with a simplified expression.

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

Let's consider the expression:

24x^2 + 36xy + 48y^2

We can see that all three terms have a common factor of 12. To factor out the GCF, we can use the distributive property as follows:

24x^2 + 36xy + 48y^2 = 12(2x^2 + 3xy + 4y^2)

In this example, we have factored out the GCF (12) from each term, leaving us with a simplified expression.

Conclusion

In conclusion, applying the distributive property to factor out the greatest common factor is a powerful technique for simplifying algebraic expressions. By identifying the GCF and distributing it over each term, we can simplify complex expressions and make them easier to work with. With practice and patience, you can master this technique and become proficient in factoring out the GCF from a wide range of expressions.

Common Mistakes to Avoid

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

  • Not identifying the GCF: Make sure to identify the greatest common factor of the terms in the expression.
  • Not distributing the GCF: Make sure to distribute the GCF over each term in the expression.
  • Not simplifying the expression: Make sure to simplify the expression by combining like terms.

Practice Problems

Here are some practice problems to help you master the technique of factoring out the GCF:

  1. Factor out the GCF from the expression: 15x + 25y
  2. Factor out the GCF from the expression: 24x^2 + 36xy + 48y^2
  3. Factor out the GCF from the expression: 18x^2 + 24xy + 30y^2

Answer Key

Here are the answers to the practice problems:

  1. 5(3x + 5y)
  2. 12(2x^2 + 3xy + 4y^2)
  3. 6(3x^2 + 4xy + 5y^2)

Final Thoughts

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

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

Q: How do I identify the GCF of an expression?

A: To identify the GCF of an expression, look for the largest number that divides each term in the expression without leaving a remainder.

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:

a(b + c) = ab + ac

Q: How do I factor out the GCF using the distributive property?

A: To factor out the GCF using the distributive property, follow these steps:

  1. Identify the GCF of the expression.
  2. Distribute the GCF over each term in the expression.
  3. Simplify the expression by combining like terms.

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 GCF
  • Not distributing the GCF
  • Not simplifying the expression

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

A: Yes, you can factor out the GCF from a complex expression. However, it may require more steps and careful attention to detail.

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 eliminate any common factors.

Q: What are some examples of expressions that can be factored using the GCF?

A: Some examples of expressions that can be factored using the GCF include:

  • 12x + 18y
  • 24x^2 + 36xy + 48y^2
  • 18x^2 + 24xy + 30y^2

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. However, be careful to distribute the negative sign correctly.

Q: How do I check my work when factoring out the GCF?

A: To check your work when factoring out the GCF, multiply the GCF by each term in the expression and simplify. If the result is the original expression, then your work is correct.

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 polynomials in computer science
  • Solving systems of equations in economics and finance

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

A: Yes, you can use a calculator to factor out the GCF. However, be careful to enter the expression correctly and follow the instructions on the calculator.

Q: How do I practice factoring out the GCF?

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

  • Use online resources and practice problems to build your skills
  • Work with a partner or tutor to get feedback and guidance
  • Practice factoring out the GCF with different types of expressions and numbers.