What Is The Product Of The Expression?${ 3x 5(2x 2 + 4x + 1) }$A. { 2x^7 + 4x^6 + X^5 $}$B. { 6x^{10} + 12x^5 + 3x^5 $}$C. { 6x^7 + 12x^6 + 3x^5 $}$D. { 3x^5 + 2x^2 + 4x + 1 $}$

Understanding the Problem

To find the product of the given expression, we need to apply the rules of algebraic multiplication. The expression is $3x^5(2x^2 + 4x + 1)$. We will use the distributive property to multiply each term inside the parentheses by the term outside the parentheses.

Applying 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. We can apply this property to our expression by multiplying each term inside the parentheses by the term outside the parentheses.

Step 1: Multiply the first term

We start by multiplying the first term inside the parentheses, which is $2x^2$, by the term outside the parentheses, which is $3x^5$. Using the rule of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the product as follows:

$3x^5 \cdot 2x^2 = 6x^{5+2} = 6x^7$

Step 2: Multiply the second term

Next, we multiply the second term inside the parentheses, which is $4x$, by the term outside the parentheses, which is $3x^5$. Using the same rule of exponents, we can simplify the product as follows:

$3x^5 \cdot 4x = 12x^{5+1} = 12x^6$

Step 3: Multiply the third term

Finally, we multiply the third term inside the parentheses, which is $1$, by the term outside the parentheses, which is $3x^5$. Since any number multiplied by 1 is itself, the product is simply:

$3x^5 \cdot 1 = 3x^5$

Combining the Terms

Now that we have multiplied each term inside the parentheses by the term outside the parentheses, we can combine the terms to find the product of the expression. The product is:

$6x^7 + 12x^6 + 3x^5$

Conclusion

The product of the expression $3x^5(2x^2 + 4x + 1)$ is $6x^7 + 12x^6 + 3x^5$. This is the correct answer.

Answer Key

The correct answer is:

A. $6x^7 + 12x^6 + 3x^5$

Comparison with Other Options

Let's compare our answer with the other options:

• Option B: $6x^{10} + 12x^5 + 3x^5$ is incorrect because the exponent of the first term is too high.
• Option C: $6x^7 + 12x^6 + 3x^5$ is the same as our answer, but it is not the only correct answer.
• Option D: $3x^5 + 2x^2 + 4x + 1$ is incorrect because it does not take into account the multiplication of the terms.

Final Answer

The final answer is:

A. $6x^7 + 12x^6 + 3x^5$

Understanding the Problem

To find the product of the given expression, we need to apply the rules of algebraic multiplication. The expression is $3x^5(2x^2 + 4x + 1)$. We will use the distributive property to multiply each term inside the parentheses by the term outside the parentheses.

Q&A

Q: What is the distributive property?

A: The distributive property is a rule of algebra that states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac. This means that we can multiply each term inside the parentheses by the term outside the parentheses.

Q: How do we apply the distributive property to the given expression?

A: We start by multiplying each term inside the parentheses by the term outside the parentheses. We can simplify the product using the rule of exponents that states $a^m \cdot a^n = a^{m+n}$.

Q: What is the product of the first term?

A: The product of the first term is $6x^7$. We get this by multiplying the first term inside the parentheses, which is $2x^2$, by the term outside the parentheses, which is $3x^5$.

Q: What is the product of the second term?

A: The product of the second term is $12x^6$. We get this by multiplying the second term inside the parentheses, which is $4x$, by the term outside the parentheses, which is $3x^5$.

Q: What is the product of the third term?

A: The product of the third term is $3x^5$. We get this by multiplying the third term inside the parentheses, which is $1$, by the term outside the parentheses, which is $3x^5$.

Q: How do we combine the terms?

A: We combine the terms by adding them together. The product of the expression is $6x^7 + 12x^6 + 3x^5$.

Q: What is the correct answer?

A: The correct answer is:

A. $6x^7 + 12x^6 + 3x^5$

Q: Why is option B incorrect?

A: Option B is incorrect because the exponent of the first term is too high. The correct exponent is $7$, not $10$.

Q: Why is option C the same as our answer?

A: Option C is the same as our answer because it also uses the distributive property to multiply each term inside the parentheses by the term outside the parentheses.

Q: Why is option D incorrect?

A: Option D is incorrect because it does not take into account the multiplication of the terms. It simply adds the terms together without multiplying them.

Conclusion

The product of the expression $3x^5(2x^2 + 4x + 1)$ is $6x^7 + 12x^6 + 3x^5$. This is the correct answer.

Final Answer

The final answer is:

A. $6x^7 + 12x^6 + 3x^5$