Use The Distributive Property To Factor Out The Greatest Common Factor For The Expression ${ 30 - 15g\$} .A. ${ 5(6 - 3g)\$} B. ${ 15(2 - G)\$} C. ${ 15(1 - G)\$} D. ${ 5(10 - 5g)\$}

Introduction

In algebra, factoring is a crucial concept that helps us simplify complex expressions and solve equations. One of the most important techniques in factoring is the distributive property, which allows us to factor out the greatest common factor (GCF) from an expression. In this article, we will explore how to use the distributive property to factor out the GCF for the expression $30 - 15g$.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra 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 distribute a single term to multiple terms inside a set of parentheses. In the context of factoring, the distributive property can be used to factor out the GCF from an expression.

Identifying the Greatest Common Factor

To factor out the GCF using the distributive property, we need to identify the greatest common factor of the terms in the expression. In the expression $30 - 15g$, the greatest common factor is $15$. This is because $15$ is the largest number that divides both $30$ and $15g$ without leaving a remainder.

Factoring Out the Greatest Common Factor

Now that we have identified the greatest common factor, we can use the distributive property to factor it out from the expression. To do this, we need to multiply the GCF by the remaining terms inside the parentheses. In this case, we have:

$$30 - 15g = 15(2 - g)$$

Here, we have factored out the GCF $15$ from the expression, leaving us with the remaining terms $2 - g$ inside the parentheses.

Checking the Answer

To check our answer, we can multiply the factored expression by the GCF to see if we get the original expression:

$$15(2 - g) = 15(2) - 15(g) = 30 - 15g$$

As we can see, the factored expression $15(2 - g)$ is equivalent to the original expression $30 - 15g$. This confirms that our answer is correct.

Conclusion

In this article, we have used the distributive property to factor out the greatest common factor from the expression $30 - 15g$. We have identified the GCF as $15$ and used the distributive property to factor it out from the expression, leaving us with the remaining terms $2 - g$ inside the parentheses. We have also checked our answer by multiplying the factored expression by the GCF to see if we get the original expression. This confirms that our answer is correct.

Answer

The correct answer is:

$$\boxed{15(2 - g)}$$

Comparison of Options

Let's compare our answer with the options provided:

A. $5(6 - 3g)$ B. $15(2 - g)$ C. $15(1 - g)$ D. $5(10 - 5g)$

As we can see, our answer $15(2 - g)$ matches option B.

Tips and Tricks

Here are some tips and tricks to help you factor out the greatest common factor using the distributive property:

• Identify the greatest common factor of the terms in the expression.
• Use the distributive property to factor out the GCF from the expression.
• Multiply the factored expression by the GCF to check your answer.
• Make sure to distribute the GCF to all the terms inside the parentheses.

By following these tips and tricks, you can master the art of factoring out the greatest common factor using the distributive property.

Practice Problems

Here are some practice problems to help you practice factoring out the greatest common factor using the distributive property:

1. Factor out the greatest common factor from the expression $24 - 12x$.
2. Factor out the greatest common factor from the expression $36 - 18y$.
3. Factor out the greatest common factor from the expression $48 - 24z$.

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about factoring out the greatest common factor using the distributive property.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra 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 distribute a single term to multiple terms inside a set of parentheses.

Q: How do I identify the greatest common factor?

A: To identify the greatest common factor, you need to find the largest number that divides all the terms in the expression without leaving a remainder. In the expression $30 - 15g$, the greatest common factor is $15$.

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

A: To factor out the greatest common factor using the distributive property, you need to multiply the GCF by the remaining terms inside the parentheses. In the expression $30 - 15g$, we can factor out the GCF $15$ as follows:

$$30 - 15g = 15(2 - g)$$

Q: How do I check my answer?

A: To check your answer, you can multiply the factored expression by the GCF to see if you get the original expression. In the expression $30 - 15g$, we can check our answer as follows:

$$15(2 - g) = 15(2) - 15(g) = 30 - 15g$$

As we can see, the factored expression $15(2 - g)$ is equivalent to the original expression $30 - 15g$. This confirms that our answer is correct.

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

A: Here are some common mistakes to avoid when factoring out the greatest common factor:

• Not identifying the greatest common factor correctly.
• Not distributing the GCF to all the terms inside the parentheses.
• Not checking the answer by multiplying the factored expression by the GCF.

Q: How can I practice factoring out the greatest common factor using the distributive property?

A: Here are some practice problems to help you practice factoring out the greatest common factor using the distributive property:

1. Factor out the greatest common factor from the expression $24 - 12x$.
2. Factor out the greatest common factor from the expression $36 - 18y$.
3. Factor out the greatest common factor from the expression $48 - 24z$.

Q: What are some real-world applications of factoring out the greatest common factor?

A: Factoring out the greatest common factor has many real-world applications, including:

• Simplifying complex expressions in algebra and calculus.
• Solving equations and inequalities in algebra and calculus.
• Finding the greatest common divisor of two or more numbers.

Conclusion

In this article, we have answered some of the most frequently asked questions about factoring out the greatest common factor using the distributive property. We have also provided some practice problems and real-world applications to help you master this technique. With practice and patience, you can become a pro at factoring out the greatest common factor using the distributive property!

Additional Resources

Here are some additional resources to help you learn more about factoring out the greatest common factor using the distributive property:

• Khan Academy: Factoring Out the Greatest Common Factor
• Mathway: Factoring Out the Greatest Common Factor
• Wolfram Alpha: Factoring Out the Greatest Common Factor

I hope this article has helped you understand how to use the distributive property to factor out the greatest common factor from an expression. With practice and patience, you can master this technique and become a pro at factoring!