find all solutions to the equation $x^3 + 16x - 3x^2 = 48$.

-4, -3, 4
-4, 3, 4
3, -4i, 4i
-3, -4i, 4i

Question

find all solutions to the equation $x^3 + 16x - 3x^2 = 48$.

-4, -3, 4
-4, 3, 4
3, -4i, 4i
-3, -4i, 4i

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Factoring by Grouping", "Complex Roots" ], "current_concepts": [ "Factoring by Grouping", "Complex Roots", "Polynomial Equations" ] } ## Rearrange the polynomial equation $$ x^3 - 3x^2 + 16x - 48 = 0 $$ ## Factor by grouping $$ \begin{aligned} x^2(x - 3) + 16(x - 3) &= 0 \ (x^2 + 16)(x - 3) &= 0 \end{aligned} $$ ## Solve for all roots $$ \begin{aligned} x - 3 = 0 &\implies x = 3 \ x^2 + 16 = 0 &\implies x^2 = -16 \implies x = \pm 4i \end{aligned} $$

(A) $-4, -3, 4$

(B) $-4, 3, 4$

(C) $3, -4i, 4i$

(D) $-3, -4i, 4i$

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

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Question

Explanation:

Rearrange the polynomial equation

Factor by grouping

Solve for all roots

Answer: