the real root of the equation $x^3 - 7x^2 + 15x - 25 = 0$ is 5. what are the nonreal roots?

$-4i, 4i$
$2 - 4i, 2 + 4i$
$1 + 2i, 1 - 2i$
$-16i, 16i$

Question

the real root of the equation $x^3 - 7x^2 + 15x - 25 = 0$ is 5. what are the nonreal roots?

$-4i, 4i$
$2 - 4i, 2 + 4i$
$1 + 2i, 1 - 2i$
$-16i, 16i$

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", "Quadratic Formula with Complex Roots" ], "new_concepts": [], "current_concepts": [ "Polynomial Division", "Quadratic Formula with Complex Roots", "Complex Roots of Polynomials" ] } ## Divide the polynomial by the known linear factor Using the *Polynomial Division* knowledge point $$ \begin{aligned} x^3 - 7x^2 + 15x - 25 &= (x - 5)(x^2 + bx + c) \ &= x^3 + (b - 5)x^2 + (c - 5b)x - 5c \ b - 5 = -7 &\implies b = -2 \ -5c = -25 &\implies c = 5 \ x^3 - 7x^2 + 15x - 25 &= (x - 5)(x^2 - 2x + 5) \end{aligned} $$ ## Solve the remaining quadratic equation Using the *Quadratic Formula with Complex Roots* knowledge point $$ \begin{aligned} x^2 - 2x + 5 &= 0 \ x &= \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(5)}}{2(1)} \ x &= \frac{2 \pm \sqrt{4 - 20}}{2} \ x &= \frac{2 \pm \sqrt{-16}}{2} \ x &= \frac{2 \pm 4i}{2} \ x &= 1 \pm 2i \end{aligned} $$

(A) $-4i, 4i$

(B) $2 - 4i, 2 + 4i$

(C) $1 + 2i, 1 - 2i$

(D) $-16i, 16i$

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

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QUESTION IMAGE

Question

Explanation:

Divide the polynomial by the known linear factor

Solve the remaining quadratic equation

Answer: