What Is The Product Of The Expression? \left(x^4\right)\left(3x^3-2\right)\left(4x^2+5x\right ]A. 12 X 9 + 15 X 8 − 8 X 6 − 10 X 5 12x^9 + 15x^8 - 8x^6 - 10x^5 12 X 9 + 15 X 8 − 8 X 6 − 10 X 5 B. 12 X 24 + 15 X 12 − 8 X 8 − 10 X 4 12x^{24} + 15x^{12} - 8x^8 - 10x^4 12 X 24 + 15 X 12 − 8 X 8 − 10 X 4 C. 12 X 9 − 10 X 5 12x^9 - 10x^5 12 X 9 − 10 X 5 D. $12x^{24} -

Understanding the Problem

To find the product of the given expression, we need to multiply the three terms together: $\left(x^4\right)\left(3x^3-2\right)\left(4x^2+5x\right)$. This involves using the distributive property to multiply each term in the first two expressions by each term in the third expression.

Distributive Property

The distributive property states that for any real numbers $a$, $b$, and $c$, the following equation holds: $a(b+c) = ab + ac$. We can use this property to expand the given expression.

Expanding the Expression

To find the product of the expression, we need to multiply each term in the first two expressions by each term in the third expression. This will result in a sum of terms, each of which is the product of three terms.

Multiplying the First Two Expressions

First, we multiply the first two expressions: $\left(x^4\right)\left(3x^3-2\right)$. Using the distributive property, we get:

$$\left(x^4\right)\left(3x^3-2\right) = x^4 \cdot 3x^3 - x^4 \cdot 2 = 3x^7 - 2x^4$$

Multiplying the Result by the Third Expression

Next, we multiply the result by the third expression: $\left(3x^7 - 2x^4\right)\left(4x^2+5x\right)$. Again, using the distributive property, we get:

$$\left(3x^7 - 2x^4\right)\left(4x^2+5x\right) = \left(3x^7\right)\left(4x^2+5x\right) - \left(2x^4\right)\left(4x^2+5x\right)$$

Expanding the Terms

Now, we expand the terms:

$$\left(3x^7\right)\left(4x^2+5x\right) = 3x^7 \cdot 4x^2 + 3x^7 \cdot 5x = 12x^9 + 15x^8$$

$$\left(2x^4\right)\left(4x^2+5x\right) = 2x^4 \cdot 4x^2 + 2x^4 \cdot 5x = 8x^6 + 10x^5$$

Combining the Terms

Finally, we combine the terms:

$$\left(3x^7 - 2x^4\right)\left(4x^2+5x\right) = 12x^9 + 15x^8 - 8x^6 - 10x^5$$

Conclusion

The product of the expression is $12x^9 + 15x^8 - 8x^6 - 10x^5$. This is the correct answer.

Discussion

The product of the expression can be found by multiplying each term in the first two expressions by each term in the third expression. This involves using the distributive property to expand the expression. The final answer is $12x^9 + 15x^8 - 8x^6 - 10x^5$.

Answer Key

The correct answer is A. $12x^9 + 15x^8 - 8x^6 - 10x^5$.

Step-by-Step Solution

1. Multiply the first two expressions: $\left(x^4\right)\left(3x^3-2\right) = 3x^7 - 2x^4$
2. Multiply the result by the third expression: $\left(3x^7 - 2x^4\right)\left(4x^2+5x\right) = \left(3x^7\right)\left(4x^2+5x\right) - \left(2x^4\right)\left(4x^2+5x\right)$
3. Expand the terms: $\left(3x^7\right)\left(4x^2+5x\right) = 12x^9 + 15x^8$ and $\left(2x^4\right)\left(4x^2+5x\right) = 8x^6 + 10x^5$
4. Combine the terms: $\left(3x^7 - 2x^4\right)\left(4x^2+5x\right) = 12x^9 + 15x^8 - 8x^6 - 10x^5$

Final Answer

The final answer is $12x^9 + 15x^8 - 8x^6 - 10x^5$.

Frequently Asked Questions

Q: What is the product of the expression $\left(x^4\right)\left(3x^3-2\right)\left(4x^2+5x\right)$?

A: The product of the expression is $12x^9 + 15x^8 - 8x^6 - 10x^5$.

Q: How do I find the product of the expression?

A: To find the product of the expression, you need to multiply each term in the first two expressions by each term in the third expression. This involves using the distributive property to expand the expression.

Q: What is the distributive property?

A: The distributive property 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 expressions by multiplying each term in one expression by each term in another expression.

Q: How do I apply the distributive property to the expression?

A: To apply the distributive property, you need to multiply each term in the first two expressions by each term in the third expression. This will result in a sum of terms, each of which is the product of three terms.

Q: What are the steps to find the product of the expression?

A: The steps to find the product of the expression are:

1. Multiply the first two expressions: $\left(x^4\right)\left(3x^3-2\right) = 3x^7 - 2x^4$
2. Multiply the result by the third expression: $\left(3x^7 - 2x^4\right)\left(4x^2+5x\right) = \left(3x^7\right)\left(4x^2+5x\right) - \left(2x^4\right)\left(4x^2+5x\right)$
3. Expand the terms: $\left(3x^7\right)\left(4x^2+5x\right) = 12x^9 + 15x^8$ and $\left(2x^4\right)\left(4x^2+5x\right) = 8x^6 + 10x^5$
4. Combine the terms: $\left(3x^7 - 2x^4\right)\left(4x^2+5x\right) = 12x^9 + 15x^8 - 8x^6 - 10x^5$

Q: What is the final answer?

A: The final answer is $12x^9 + 15x^8 - 8x^6 - 10x^5$.

Common Mistakes

Mistake 1: Not using the distributive property

A: One common mistake is not using the distributive property to expand the expression. This can lead to incorrect results.

Mistake 2: Not multiplying each term correctly

A: Another common mistake is not multiplying each term correctly. This can also lead to incorrect results.

Mistake 3: Not combining the terms correctly

A: A third common mistake is not combining the terms correctly. This can also lead to incorrect results.

Tips and Tricks

Tip 1: Use the distributive property to expand the expression

A: To find the product of the expression, use the distributive property to expand the expression.

Tip 2: Multiply each term correctly

A: To find the product of the expression, multiply each term correctly.

Tip 3: Combine the terms correctly

A: To find the product of the expression, combine the terms correctly.

Conclusion

The product of the expression $\left(x^4\right)\left(3x^3-2\right)\left(4x^2+5x\right)$ is $12x^9 + 15x^8 - 8x^6 - 10x^5$. To find the product of the expression, use the distributive property to expand the expression, multiply each term correctly, and combine the terms correctly.