Question

what is the product?

(3x^5(2x^2 + 4x + 1))

• (2x^7 + 4x^6 + x^5)
• (6x^{10} + 12x^5 + 3x^5)
• (6x^7 + 12x^6 + 3x^5)
• (3x^5 + 2x^2 + 4x + 1)

Explanation:

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</pre_analysis>

<reasoning>

Distribute the monomial to each term

\[
3x^5(2x^2 + 4x + 1) = (3x^5 \cdot 2x^2) + (3x^5 \cdot 4x) + (3x^5 \cdot 1)
\]

Apply exponent rules to multiply terms

\[

$$\begin{aligned} 3x^5 \cdot 2x^2 &= (3 \cdot 2)x^{5+2} = 6x^7 \\ 3x^5 \cdot 4x &= (3 \cdot 4)x^{5+1} = 12x^6 \\ 3x^5 \cdot 1 &= 3x^5 \end{aligned}$$

\]

Combine the simplified terms

\[
6x^7 + 12x^6 + 3x^5
\]
</reasoning>

<answer>
<mcq-option>(A) \(2x^7 + 4x^6 + x^5\)</mcq-option>
<mcq-option>(B) \(6x^{10} + 12x^5 + 3x^5\)</mcq-option>
<mcq-correct>(C) \(6x^7 + 12x^6 + 3x^5\)</mcq-correct>
<mcq-option>(D) \(3x^5 + 2x^2 + 4x + 1\)</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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"Mathematics",
"Algebra",
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]
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</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
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</pre_analysis>

<reasoning>

Distribute the monomial to each term

\[
3x^5(2x^2 + 4x + 1) = (3x^5 \cdot 2x^2) + (3x^5 \cdot 4x) + (3x^5 \cdot 1)
\]

Apply exponent rules to multiply terms

\[

$$\begin{aligned} 3x^5 \cdot 2x^2 &= (3 \cdot 2)x^{5+2} = 6x^7 \\ 3x^5 \cdot 4x &= (3 \cdot 4)x^{5+1} = 12x^6 \\ 3x^5 \cdot 1 &= 3x^5 \end{aligned}$$

\]

Combine the simplified terms

\[
6x^7 + 12x^6 + 3x^5
\]
</reasoning>

<answer>
<mcq-option>(A) \(2x^7 + 4x^6 + x^5\)</mcq-option>
<mcq-option>(B) \(6x^{10} + 12x^5 + 3x^5\)</mcq-option>
<mcq-correct>(C) \(6x^7 + 12x^6 + 3x^5\)</mcq-correct>
<mcq-option>(D) \(3x^5 + 2x^2 + 4x + 1\)</mcq-option>
</answer>

<post_analysis>
{
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Polynomial Multiplication"
]
}
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