the real solutions to the equation $x^4 - 5x^3 - 61x = 150$ are shown in the graph. what are the other solutions?

$-3 + 4i, -3 - 4i$
$3 + 4i, 3 - 4i$
$-6 + 8i, -6 - 8i$
$6 + 4i, 6 - 4i$

Question

the real solutions to the equation $x^4 - 5x^3 - 61x = 150$ are shown in the graph. what are the other solutions?

$-3 + 4i, -3 - 4i$
$3 + 4i, 3 - 4i$
$-6 + 8i, -6 - 8i$
$6 + 4i, 6 - 4i$

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": [ "Rational Roots from Graph", "Polynomial Division", "Quadratic Formula with Complex Roots" ], "new_concepts": [], "current_concepts": [ "Rational Roots from Graph", "Polynomial Division", "Quadratic Formula with Complex Roots", "Complex Roots of Polynomials" ] } ## Identify real roots from the graph Using the *Rational Roots from Graph* knowledge point $$ \begin{aligned} &x_1 = -3\ &x_2 = 2 \end{aligned} $$ ## Perform polynomial division by the known real roots Using the *Polynomial Division* knowledge point $$ \begin{aligned} P(x) &= x^4 - 5x^3 - 11x^2 - 61x - 150\ (x+3)(x-2) &= x^2 + x - 6\ \frac{P(x)}{x^2+x-6} &= x^2 - 6x + 25 \end{aligned} $$ ## Solve the remaining quadratic equation Using the *Quadratic Formula with Complex Roots* knowledge point $$ \begin{aligned} &x^2 - 6x + 25 = 0\ &x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(25)}}{2}\ &x = \frac{6 \pm \sqrt{36 - 100}}{2}\ &x = \frac{6 \pm \sqrt{-64}}{2}\ &x = 3 \pm 4i \end{aligned} $$

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

(B) $3 + 4i, 3 - 4i$

(C) $-6 + 8i, -6 - 8i$

(D) $6 + 4i, 6 - 4i$

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

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QUESTION IMAGE

Question

Explanation:

Identify real roots from the graph

Perform polynomial division by the known real roots

Solve the remaining quadratic equation

Answer: