Question

what is the factorization of (1,000x^6 - 27)?

\\( (10x^2 - 3)(100x^2 + 30x^2 + 9) \\)
\\( (10x^2 - 3)(100x^4 + 30x^2 + 9) \\)
\\( (10x^3 - 3)(100x^2 + 30x^3 + 9) \\)
\\( (10x^3 - 3)(100x^6 + 30x^3 + 9) \\)

Explanation:

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</pre_analysis>

<reasoning>

Rewrite as a difference of cubes

\[
1,000x^6 - 27 = (10x^2)^3 - 3^3
\]

Apply the difference of cubes formula

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
\[

$$\begin{aligned} a &= 10x^2 \\ b &= 3 \end{aligned}$$

\]
\[
(10x^2 - 3)((10x^2)^2 + (10x^2)(3) + 3^2)
\]

Simplify the terms

\[
(10x^2 - 3)(100x^4 + 30x^2 + 9)
\]
</reasoning>

<answer>
<mcq-option>(A) \((10x^2 - 3)(100x^2 + 30x^2 + 9)\)</mcq-option>
<mcq-correct>(B) \((10x^2 - 3)(100x^4 + 30x^2 + 9)\)</mcq-correct>
<mcq-option>(C) \((10x^3 - 3)(100x^2 + 30x^3 + 9)\)</mcq-option>
<mcq-option>(D) \((10x^3 - 3)(100x^6 + 30x^3 + 9)\)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Difference of Cubes"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
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"skills_matched": [
"step_cot",
"multiple_choice_formatter",
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"concepts_used": [
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"new_concepts": [],
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</pre_analysis>

<reasoning>

Rewrite as a difference of cubes

\[
1,000x^6 - 27 = (10x^2)^3 - 3^3
\]

Apply the difference of cubes formula

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
\[

$$\begin{aligned} a &= 10x^2 \\ b &= 3 \end{aligned}$$

\]
\[
(10x^2 - 3)((10x^2)^2 + (10x^2)(3) + 3^2)
\]

Simplify the terms

\[
(10x^2 - 3)(100x^4 + 30x^2 + 9)
\]
</reasoning>

<answer>
<mcq-option>(A) \((10x^2 - 3)(100x^2 + 30x^2 + 9)\)</mcq-option>
<mcq-correct>(B) \((10x^2 - 3)(100x^4 + 30x^2 + 9)\)</mcq-correct>
<mcq-option>(C) \((10x^3 - 3)(100x^2 + 30x^3 + 9)\)</mcq-option>
<mcq-option>(D) \((10x^3 - 3)(100x^6 + 30x^3 + 9)\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Difference of Cubes"
]
}
</post_analysis>