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{ "quality": "incomplete", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Polynomial Multiplication" ], "current_concepts": [ "Polynomial Multiplication", "Algebraic Simplification" ] } ## Identify the factors from the options $$ \begin{aligned} &\text{Let the product be } P(x) = 2x^3 + Bx^2 - x - 6\ &\text{Since } P(2) = 16 + 4B - 2 - 6 = 8 + 4B \text{ and } P(-1) = -2 + B + 1 - 6 = B - 7\ &\text{Testing standard binomial factors for } 2x^3 + 5x^2 - x - 6:\ &(2x + 3)(x + 2)(x - 1) = (2x^2 + 7x + 6)(x - 1) = 2x^3 + 5x^2 - x - 6 \end{aligned} $$ ## Verify the polynomial multiplication $$ \begin{aligned} &(2x^2 + 7x + 6)(x - 1) \ &= 2x^2(x - 1) + 7x(x - 1) + 6(x - 1) \ &= 2x^3 - 2x^2 + 7x^2 - 7x + 6x - 6 \ &= 2x^3 + 5x^2 - x - 6 \end{aligned} $$

(A) $2x^3 + 6x^2 - x - 6$

(B) $2x^3 + x^2 - x - 6$

(C) $2x^3 + 5x^2 - x - 6$

(D) $2x^3 - x^2 - 7x - 6$

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

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Explanation:

Identify the factors from the options

Verify the polynomial multiplication

Answer: