Question

what is the factored form of \\(x^3 + 125\\)?

\\((x^3 + 5)(x^6 - 5x^3 + 25)\\)
\\((x^3 - 5)(x^6 + 5x^3 + 25)\\)
\\((x + 5)(x^2 - 5x + 25)\\)
\\((x - 5)(x^2 + 5x + 25)\\)

Explanation:

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<reasoning>

Identify the terms as perfect cubes

Using the Sum of Cubes knowledge point
\[

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

\]

Apply the sum of cubes formula

Using the Sum of Cubes knowledge point
\[

$$\begin{aligned} a^3 + b^3 &= (a + b)(a^2 - ab + b^2) \\ x^3 + 5^3 &= (x + 5)(x^2 - 5x + 5^2) \\ &= (x + 5)(x^2 - 5x + 25) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((x^3 + 5)(x^6 - 5x^3 + 25)\)</mcq-option>
<mcq-option>(B) \((x^3 - 5)(x^6 + 5x^3 + 25)\)</mcq-option>
<mcq-correct>(C) \((x + 5)(x^2 - 5x + 25)\)</mcq-correct>
<mcq-option>(D) \((x - 5)(x^2 + 5x + 25)\)</mcq-option>
</answer>

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

Answer:

<pre_analysis>
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"step_cot",
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</pre_analysis>

<reasoning>

Identify the terms as perfect cubes

Using the Sum of Cubes knowledge point
\[

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

\]

Apply the sum of cubes formula

Using the Sum of Cubes knowledge point
\[

$$\begin{aligned} a^3 + b^3 &= (a + b)(a^2 - ab + b^2) \\ x^3 + 5^3 &= (x + 5)(x^2 - 5x + 5^2) \\ &= (x + 5)(x^2 - 5x + 25) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((x^3 + 5)(x^6 - 5x^3 + 25)\)</mcq-option>
<mcq-option>(B) \((x^3 - 5)(x^6 + 5x^3 + 25)\)</mcq-option>
<mcq-correct>(C) \((x + 5)(x^2 - 5x + 25)\)</mcq-correct>
<mcq-option>(D) \((x - 5)(x^2 + 5x + 25)\)</mcq-option>
</answer>

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