Question

factor \\(64 - x^{15}\\).

\\((4 - x^3)(16 + 4x^3 + x^3)\\)
\\((4 - x^3)(16 + 4x^3 + x^6)\\)
\\((4 - x^5)(16 + 4x^5 + x^5)\\)
\\((4 - x^5)(16 + 4x^5 + x^{10})\\)

Explanation:

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<reasoning>

Identify the terms as perfect cubes

\[

$$\begin{aligned} 64 &= 4^3 \\ x^{15} &= (x^5)^3 \end{aligned}$$

\]

Apply the difference of cubes formula

\[

$$\begin{aligned} a^3 - b^3 &= (a - b)(a^2 + ab + b^2) \\ \text{Let } a = 4 &\text{ and } b = x^5 \end{aligned}$$

\]

Substitute and simplify the expression

\[

$$\begin{aligned} 64 - x^{15} &= (4 - x^5)(4^2 + 4(x^5) + (x^5)^2) \\ &= (4 - x^5)(16 + 4x^5 + x^{10}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((4-x^3)(16+4x^3+x^3)\)</mcq-option>
<mcq-option>(B) \((4-x^3)(16+4x^3+x^6)\)</mcq-option>
<mcq-option>(C) \((4-x^5)(16+4x^5+x^5)\)</mcq-option>
<mcq-correct>(D) \((4-x^5)(16+4x^5+x^{10})\)</mcq-correct>
</answer>

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Answer:

<pre_analysis>
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<reasoning>

Identify the terms as perfect cubes

\[

$$\begin{aligned} 64 &= 4^3 \\ x^{15} &= (x^5)^3 \end{aligned}$$

\]

Apply the difference of cubes formula

\[

$$\begin{aligned} a^3 - b^3 &= (a - b)(a^2 + ab + b^2) \\ \text{Let } a = 4 &\text{ and } b = x^5 \end{aligned}$$

\]

Substitute and simplify the expression

\[

$$\begin{aligned} 64 - x^{15} &= (4 - x^5)(4^2 + 4(x^5) + (x^5)^2) \\ &= (4 - x^5)(16 + 4x^5 + x^{10}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((4-x^3)(16+4x^3+x^3)\)</mcq-option>
<mcq-option>(B) \((4-x^3)(16+4x^3+x^6)\)</mcq-option>
<mcq-option>(C) \((4-x^5)(16+4x^5+x^5)\)</mcq-option>
<mcq-correct>(D) \((4-x^5)(16+4x^5+x^{10})\)</mcq-correct>
</answer>

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