using the distributive property to find the product $(y-4x)(y^2+4y+16)$ results in a polynomial of the form $y^3+4y^2+ay-4xy^2-axy-64x$. what is the value of $a$ in the polynomial?

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Question

using the distributive property to find the product $(y-4x)(y^2+4y+16)$ results in a polynomial of the form $y^3+4y^2+ay-4xy^2-axy-64x$. what is the value of $a$ in the polynomial?

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8
16
32

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Multiplication", "Algebraic Simplification" ], "new_concepts": [ "Comparing Polynomial Coefficients" ], "current_concepts": [ "Polynomial Multiplication", "Algebraic Simplification", "Comparing Polynomial Coefficients" ] } ## Expand the product using the distributive property $$ \begin{aligned} (y - 4x)(y^2 + 4y + 16) &= y(y^2 + 4y + 16) - 4x(y^2 + 4y + 16) \ &= y^3 + 4y^2 + 16y - 4xy^2 - 16xy - 64x \end{aligned} $$ ## Compare the expanded expression with the given form $$ \begin{aligned} \text{Given form: } & y^3 + 4y^2 + ay - 4xy^2 - axy - 64x \ \text{Expanded form: } & y^3 + 4y^2 + 16y - 4xy^2 - 16xy - 64x \end{aligned} $$ ## Solve for the coefficient a $$ a = 16 $$

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{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Multiplication" ] }

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QUESTION IMAGE

Question

Explanation:

Expand the product using the distributive property

Compare the expanded expression with the given form

Solve for the coefficient a

Answer: