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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Polynomial Equation Solving", "Complex Roots" ], "current_concepts": [ "Roots of Polynomials", "Quadratic Formula with Complex Roots", "Complex Roots of Polynomials" ] } ## Simplify the polynomial equation $$ \begin{aligned} 3x^3 - 6x^2 + 30x - 20 &= -20 \ 3x^3 - 6x^2 + 30x &= 0 \ 3x(x^2 - 2x + 10) &= 0 \end{aligned} $$ ## Solve for the real root $$ \begin{aligned} 3x &= 0 \ x &= 0 \end{aligned} $$ ## Solve for the complex roots $$ \begin{aligned} x^2 - 2x + 10 &= 0 \ x &= \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(10)}}{2(1)} \ x &= \frac{2 \pm \sqrt{4 - 40}}{2} \ x &= \frac{2 \pm \sqrt{-36}}{2} \ x &= \frac{2 \pm 6i}{2} \ x &= 1 \pm 3i \end{aligned} $$

(A) 0, 1 + 3i, 1 - 3i

(B) 0, 2 + 6i, 2 - 6i

(C) 0, -1 + 3i, -1 - 3i

(D) 0, -2 + 6i, -2 - 6i

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Complex Roots of Polynomials" ] }

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Explanation:

Simplify the polynomial equation

Solve for the real root

Solve for the complex roots

Answer: