if the real roots of the equation $3x^4 - 6x^3 + 3x^2 - 54x - 216 = 0$ are $-2$ and $4$, what are the nonreal roots?

$-3i, 3i$
$-3i\sqrt{3}, 3i\sqrt{3}$
$-9i, 9i$
$-6i\sqrt{6}, 6i\sqrt{6}$

Question

if the real roots of the equation $3x^4 - 6x^3 + 3x^2 - 54x - 216 = 0$ are $-2$ and $4$, what are the nonreal roots?

$-3i, 3i$
$-3i\sqrt{3}, 3i\sqrt{3}$
$-9i, 9i$
$-6i\sqrt{6}, 6i\sqrt{6}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Polynomial Division", "Roots of Polynomials" ], "new_concepts": [ "Complex Roots of Polynomials" ], "current_concepts": [ "Polynomial Division", "Roots of Polynomials", "Complex Roots of Polynomials" ] } ## Factor out the common constant factor Using the *Polynomial Division* knowledge point $$ \begin{aligned} &3x^4 - 6x^3 + 3x^2 - 54x - 216 = 0 \ &3(x^4 - 2x^3 + x^2 - 18x - 72) = 0 \ &x^4 - 2x^3 + x^2 - 18x - 72 = 0 \end{aligned} $$ ## Construct the quadratic factor from the real roots Using the *Roots of Polynomials* knowledge point $$ \begin{aligned} &(x - (-2))(x - 4) = (x + 2)(x - 4) \ &(x + 2)(x - 4) = x^2 - 2x - 8 \end{aligned} $$ ## Divide the polynomial to find the remaining quadratic factor Using the *Polynomial Division* knowledge point $$ \begin{aligned} &\frac{x^4 - 2x^3 + x^2 - 18x - 72}{x^2 - 2x - 8} \ &= x^2 + 9 \end{aligned} $$ ## Solve for the nonreal roots Using the *Roots of Polynomials* knowledge point $$ \begin{aligned} &x^2 + 9 = 0 \ &x^2 = -9 \ &x = \pm 3i \end{aligned} $$

(A) $-3i, 3i$

(B) $-3i\sqrt{3}, 3i\sqrt{3}$

(C) $-9i, 9i$

(D) $-6i\sqrt{6}, 6i\sqrt{6}$

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

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QUESTION IMAGE

Question

Explanation:

Factor out the common constant factor

Construct the quadratic factor from the real roots

Divide the polynomial to find the remaining quadratic factor

Solve for the nonreal roots

Answer: