Question

rewrite the following expression in expanded form and simplify completely.

\\(3x - y)^4\\

Explanation:

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"Binomial Expansion"
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</pre_analysis>

<reasoning>

Apply the Binomial Theorem formula

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
\[
a = 3x, \quad b = -y, \quad n = 4
\]

Expand the terms using binomial coefficients

\[
\binom{4}{0}(3x)^4(-y)^0 + \binom{4}{1}(3x)^3(-y)^1 + \binom{4}{2}(3x)^2(-y)^2 + \binom{4}{3}(3x)^1(-y)^3 + \binom{4}{4}(3x)^0(-y)^4
\]
\[
1(81x^4)(1) + 4(27x^3)(-y) + 6(9x^2)(y^2) + 4(3x)(-y^3) + 1(1)(y^4)
\]

Simplify the expanded expression

\[
81x^4 - 108x^3y + 54x^2y^2 - 12xy^3 + y^4
\]
</reasoning>

<answer>
Rewrite the following expression in expanded form and simplify completely.
\[(3x - y)^4 =\] <blank>\(81x^4 - 108x^3y + 54x^2y^2 - 12xy^3 + y^4\)</blank>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Binomial Expansion"
]
}
</post_analysis>

Answer:

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"Binomial Expansion"
],
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</pre_analysis>

<reasoning>

Apply the Binomial Theorem formula

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
\[
a = 3x, \quad b = -y, \quad n = 4
\]

Expand the terms using binomial coefficients

\[
\binom{4}{0}(3x)^4(-y)^0 + \binom{4}{1}(3x)^3(-y)^1 + \binom{4}{2}(3x)^2(-y)^2 + \binom{4}{3}(3x)^1(-y)^3 + \binom{4}{4}(3x)^0(-y)^4
\]
\[
1(81x^4)(1) + 4(27x^3)(-y) + 6(9x^2)(y^2) + 4(3x)(-y^3) + 1(1)(y^4)
\]

Simplify the expanded expression

\[
81x^4 - 108x^3y + 54x^2y^2 - 12xy^3 + y^4
\]
</reasoning>

<answer>
Rewrite the following expression in expanded form and simplify completely.
\[(3x - y)^4 =\] <blank>\(81x^4 - 108x^3y + 54x^2y^2 - 12xy^3 + y^4\)</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Binomial Expansion"
]
}
</post_analysis>