What Is The Product Of 3 X ( X 2 + 4 3x(x^2+4 3 X ( X 2 + 4 ]?A. X 2 + 3 X + 4 X^2 + 3x + 4 X 2 + 3 X + 4 B. 3 X 3 + 4 3x^3 + 4 3 X 3 + 4 C. 3 X 3 + 12 X 3x^3 + 12x 3 X 3 + 12 X D. 3 X 2 + 12 X 3x^2 + 12x 3 X 2 + 12 X

$$

Understanding the Problem

To find the product of $3x(x^2+4)$, we need to apply the distributive property of multiplication over addition. This means that we will multiply each term inside the parentheses by the term outside the parentheses, which is $3x$. The distributive property is a fundamental concept in algebra that allows us to expand expressions and simplify them.

Applying the Distributive Property

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

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

In our case, we have:

$3x(x^2+4) = 3x \cdot x^2 + 3x \cdot 4$

Multiplying the Terms

Now, we will multiply the terms inside the parentheses by the term outside the parentheses.

$3x \cdot x^2 = 3x^3$

$3x \cdot 4 = 12x$

Combining the Terms

Now that we have multiplied the terms, we can combine them to get the final product.

$3x^3 + 12x$

Comparing the Options

Let's compare our result with the options given:

A. $x^2 + 3x + 4$

B. $3x^3 + 4$

C. $3x^3 + 12x$

D. $3x^2 + 12x$

Our result matches option C, which is $3x^3 + 12x$.

Conclusion

In this article, we have applied the distributive property of multiplication over addition to find the product of $3x(x^2+4)$. We have multiplied the terms inside the parentheses by the term outside the parentheses and combined them to get the final product. Our result matches option C, which is $3x^3 + 12x$.

Frequently Asked Questions

• What is the distributive property of multiplication over addition? The distributive property of multiplication over addition states that for any real numbers $a$, $b$, and $c$, we have: $a(b+c) = ab + ac$
• How do I apply the distributive property to find the product of an expression? To apply the distributive property, you need to multiply each term inside the parentheses by the term outside the parentheses.
• What is the final product of $3x(x^2+4)$? The final product of $3x(x^2+4)$ is $3x^3 + 12x$.

Step-by-Step Solution

1. Apply the distributive property to the expression $3x(x^2+4)$.
2. Multiply each term inside the parentheses by the term outside the parentheses.
3. Combine the terms to get the final product.

Common Mistakes

• Not applying the distributive property correctly.
• Not multiplying each term inside the parentheses by the term outside the parentheses.
• Not combining the terms correctly.

Real-World Applications

The distributive property of multiplication over addition has many real-world applications, such as:

• Simplifying algebraic expressions.
• Solving equations and inequalities.
• Finding the area and perimeter of geometric shapes.

Conclusion

In this article, we have applied the distributive property of multiplication over addition to find the product of $3x(x^2+4)$. We have multiplied the terms inside the parentheses by the term outside the parentheses and combined them to get the final product. Our result matches option C, which is $3x^3 + 12x$.

$$

Q&A: Product of $3x(x^2+4)$

Q: What is the product of $3x(x^2+4)$?

A: The product of $3x(x^2+4)$ is $3x^3 + 12x$.

Q: How do I apply the distributive property to find the product of an expression?

A: To apply the distributive property, you need to multiply each term inside the parentheses by the term outside the parentheses.

Q: What is the distributive property of multiplication over addition?

A: The distributive property of multiplication over addition states that for any real numbers $a$, $b$, and $c$, we have: $a(b+c) = ab + ac$

Q: How do I simplify an algebraic expression using the distributive property?

A: To simplify an algebraic expression using the distributive property, you need to multiply each term inside the parentheses by the term outside the parentheses and combine the 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 applying the distributive property correctly.
• Not multiplying each term inside the parentheses by the term outside the parentheses.
• Not combining the terms correctly.

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

A: Some real-world applications of the distributive property include:

• Simplifying algebraic expressions.
• Solving equations and inequalities.
• Finding the area and perimeter of geometric shapes.

Q: How do I check my work when applying the distributive property?

A: To check your work when applying the distributive property, you can:

• Multiply each term inside the parentheses by the term outside the parentheses.
• Combine the terms to get the final product.
• Compare your result with the correct answer.

Q: What is the final product of $3x(x^2+4)$?

A: The final product of $3x(x^2+4)$ is $3x^3 + 12x$.

Q: Can I use the distributive property to simplify expressions with more than two terms?

A: Yes, you can use the distributive property to simplify expressions with more than two terms. Simply multiply each term inside the parentheses by the term outside the parentheses and combine the terms.

Q: How do I apply the distributive property to expressions with variables and constants?

A: To apply the distributive property to expressions with variables and constants, you need to multiply each term inside the parentheses by the term outside the parentheses and combine the terms.

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

A: Some tips for applying the distributive property correctly include:

• Read the expression carefully and identify the terms inside the parentheses.
• Multiply each term inside the parentheses by the term outside the parentheses.
• Combine the terms to get the final product.
• Check your work by comparing your result with the correct answer.

Conclusion

In this article, we have answered some frequently asked questions about the product of $3x(x^2+4)$. We have discussed the distributive property of multiplication over addition, how to apply it to find the product of an expression, and some common mistakes to avoid. We have also provided some real-world applications of the distributive property and some tips for applying it correctly.