Question

factoring a sum of cubes

factor \\(125x^9 + 64\\).

\\((5x^3 - 4)(25x^6 + 20x^3 + 16)\\)
\\((5x^3 - 4)(25x^3 + 20x^3 + 16)\\)
\\((5x^3 + 4)(25x^6 - 20x^3 + 16)\\)
\\((5x^3 + 4)(25x^3 - 20x^3 + 16)\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Sum of Cubes"
],
"new_concepts": [],
"current_concepts": [
"Sum of Cubes",
"Factoring Polynomials"
]
}
</pre_analysis>

<reasoning>

Rewrite terms as perfect cubes

Using the Sum of Cubes knowledge point
\[

$$\begin{aligned} 125x^9 &= (5x^3)^3 \\ 64 &= 4^3 \end{aligned}$$

\]

Apply the sum of cubes formula

Using the Sum of Cubes knowledge point
\[

$$\begin{aligned} a^3 + b^3 &= (a + b)(a^2 - ab + b^2) \\ a &= 5x^3, \quad b = 4 \\ 125x^9 + 64 &= (5x^3 + 4)((5x^3)^2 - (5x^3)(4) + 4^2) \end{aligned}$$

\]

Simplify the factored expression

Using the Sum of Cubes knowledge point
\[

$$\begin{aligned} (5x^3)^2 &= 25x^6 \\ (5x^3)(4) &= 20x^3 \\ 4^2 &= 16 \\ 125x^9 + 64 &= (5x^3 + 4)(25x^6 - 20x^3 + 16) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((5x^3 - 4)(25x^6 + 20x^3 + 16)\)</mcq-option>
<mcq-option>(B) \((5x^3 - 4)(25x^3 + 20x^3 + 16)\)</mcq-option>
<mcq-correct>(C) \((5x^3 + 4)(25x^6 - 20x^3 + 16)\)</mcq-correct>
<mcq-option>(D) \((5x^3 + 4)(25x^3 - 20x^3 + 16)\)</mcq-option>
</answer>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Sum of Cubes"
],
"new_concepts": [],
"current_concepts": [
"Sum of Cubes",
"Factoring Polynomials"
]
}
</pre_analysis>

<reasoning>

Rewrite terms as perfect cubes

Using the Sum of Cubes knowledge point
\[

$$\begin{aligned} 125x^9 &= (5x^3)^3 \\ 64 &= 4^3 \end{aligned}$$

\]

Apply the sum of cubes formula

Using the Sum of Cubes knowledge point
\[

$$\begin{aligned} a^3 + b^3 &= (a + b)(a^2 - ab + b^2) \\ a &= 5x^3, \quad b = 4 \\ 125x^9 + 64 &= (5x^3 + 4)((5x^3)^2 - (5x^3)(4) + 4^2) \end{aligned}$$

\]

Simplify the factored expression

Using the Sum of Cubes knowledge point
\[

$$\begin{aligned} (5x^3)^2 &= 25x^6 \\ (5x^3)(4) &= 20x^3 \\ 4^2 &= 16 \\ 125x^9 + 64 &= (5x^3 + 4)(25x^6 - 20x^3 + 16) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((5x^3 - 4)(25x^6 + 20x^3 + 16)\)</mcq-option>
<mcq-option>(B) \((5x^3 - 4)(25x^3 + 20x^3 + 16)\)</mcq-option>
<mcq-correct>(C) \((5x^3 + 4)(25x^6 - 20x^3 + 16)\)</mcq-correct>
<mcq-option>(D) \((5x^3 + 4)(25x^3 - 20x^3 + 16)\)</mcq-option>
</answer>

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