Question

express the following polynomial in expanded form and simplify completely.
\\p(x) = (4x - 1)^3\\

Explanation:

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</pre_analysis>

<reasoning>

Apply the binomial expansion formula for a cube

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

Substitute the terms into the formula

\[

$$\begin{aligned} a &= 4x \\ b &= 1 \\ p(x) &= (4x)^3 - 3(4x)^2(1) + 3(4x)(1)^2 - (1)^3 \end{aligned}$$

\]

Simplify each term completely

\[

$$\begin{aligned} (4x)^3 &= 64x^3 \\ -3(16x^2)(1) &= -48x^2 \\ 3(4x)(1) &= 12x \\ -(1)^3 &= -1 \\ p(x) &= 64x^3 - 48x^2 + 12x - 1 \end{aligned}$$

\]
</reasoning>

<answer>
\(p(x) = 64x^3 - 48x^2 + 12x - 1\)
</answer>

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"subject": "Mathematics",
"question_type": "Free Response",
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]
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</post_analysis>

Answer:

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"step_cot",
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],
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</pre_analysis>

<reasoning>

Apply the binomial expansion formula for a cube

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

Substitute the terms into the formula

\[

$$\begin{aligned} a &= 4x \\ b &= 1 \\ p(x) &= (4x)^3 - 3(4x)^2(1) + 3(4x)(1)^2 - (1)^3 \end{aligned}$$

\]

Simplify each term completely

\[

$$\begin{aligned} (4x)^3 &= 64x^3 \\ -3(16x^2)(1) &= -48x^2 \\ 3(4x)(1) &= 12x \\ -(1)^3 &= -1 \\ p(x) &= 64x^3 - 48x^2 + 12x - 1 \end{aligned}$$

\]
</reasoning>

<answer>
\(p(x) = 64x^3 - 48x^2 + 12x - 1\)
</answer>

<post_analysis>
{
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"question_type": "Free Response",
"knowledge_point": [
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]
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