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

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Binomial Theorem", "Binomial Expansion" ], "new_concepts": [], "current_concepts": [ "Binomial Theorem", "Binomial Expansion", "Polynomial Expansion" ] } ## 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 $$ \begin{aligned} (4x)^3 &= 64x^3 \ -3(16x^2)(1) &= -48x^2 \ 3(4x)(1) &= 12x \ -(1)^3 &= -1 \ p(x) &= 64x^3 - 48x^2 + 12x - 1 \end{aligned} $$ $p(x) = 64x^3 - 48x^2 + 12x - 1$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Binomial Expansion" ] }

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Explanation:

Apply the binomial expansion formula for a cube

Substitute the terms into the formula

Simplify each term completely

Answer: