The Distributive Property Can Be Applied To Which Expression To Factor $12x^3 - 9x^2 + 4x - 3$?A. $4x(3x^2 + 1) - 3(3x^2 - 1$\] B. $3x(4x - 3) - (3 + 4x$\] C. $3x^2(4x - 3) + 1(4x - 3$\]

Introduction

The distributive property is a fundamental concept in algebra that allows us to factor expressions by distributing a common factor to each term. In this article, we will explore how the distributive property can be applied to factor a given expression, and we will examine the correct answer among the options provided.

Understanding the Distributive Property

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

a(b + c) = ab + ac

This property can be extended to expressions with variables, such as:

a(bx + cx) = abx + acx

The distributive property is a powerful tool for factoring expressions, as it allows us to break down complex expressions into simpler ones.

Factoring the Given Expression

The given expression is:

$$12x^3 - 9x^2 + 4x - 3$$

To factor this expression using the distributive property, we need to find a common factor that can be distributed to each term. Let's examine the options provided:

Option A: $4x(3x^2 + 1) - 3(3x^2 - 1)$

To factor the given expression using the distributive property, we can try to find a common factor that can be distributed to each term. In this case, we can see that the first two terms have a common factor of $4x$, and the last two terms have a common factor of $-3$. Therefore, we can write:

$$12x^3 - 9x^2 + 4x - 3 = 4x(3x^2 + 1) - 3(3x^2 - 1)$$

This expression can be further simplified by distributing the common factors:

$$4x(3x^2 + 1) - 3(3x^2 - 1) = 12x^3 + 4x - 9x^2 + 3$$

However, this expression is not equal to the original expression. Therefore, option A is not the correct answer.

Option B: $3x(4x - 3) - (3 + 4x)$

To factor the given expression using the distributive property, we can try to find a common factor that can be distributed to each term. In this case, we can see that the first two terms have a common factor of $3x$, and the last term has a common factor of $-1$. Therefore, we can write:

$$12x^3 - 9x^2 + 4x - 3 = 3x(4x - 3) - (3 + 4x)$$

This expression can be further simplified by distributing the common factors:

$$3x(4x - 3) - (3 + 4x) = 12x^3 - 9x^2 + 4x - 3$$

This expression is equal to the original expression. Therefore, option B is the correct answer.

Option C: $3x^2(4x - 3) + 1(4x - 3)$

To factor the given expression using the distributive property, we can try to find a common factor that can be distributed to each term. In this case, we can see that the first two terms have a common factor of $3x^2$, and the last term has a common factor of $1$. Therefore, we can write:

$$12x^3 - 9x^2 + 4x - 3 = 3x^2(4x - 3) + 1(4x - 3)$$

This expression can be further simplified by distributing the common factors:

$$3x^2(4x - 3) + 1(4x - 3) = 12x^3 - 9x^2 + 4x - 3$$

However, this expression is not equal to the original expression. Therefore, option C is not the correct answer.

Conclusion

In this article, we have explored how the distributive property can be applied to factor a given expression. We have examined three options and found that option B is the correct answer. The distributive property is a powerful tool for factoring expressions, and it can be used to break down complex expressions into simpler ones.

Final Answer

The final answer is option B: $3x(4x - 3) - (3 + 4x)$.

Additional Resources

For more information on the distributive property and factoring expressions, please refer to the following resources:

• Khan Academy: Distributive Property
• Mathway: Factoring Expressions
• Wolfram Alpha: Distributive Property

FAQs

Q: What is the distributive property? A: The distributive property is a fundamental concept in algebra that allows us to factor expressions by distributing a common factor to each term.

Q: How can I apply the distributive property to factor an expression? A: To apply the distributive property, you need to find a common factor that can be distributed to each term. You can then simplify the expression by distributing the common factors.

Q: What is the correct answer among the options provided? A: The correct answer is option B: $3x(4x - 3) - (3 + 4x)$.