for the polynomial, $p(x) = x^3 - 13x^2 + 72x - 160$, 5 is one of the zeros. find all remaining zeros.

the remaining zeros are .
(simplify your answer. type your answer in the form $a + bi$. use a comma to separate answers as needed.)

Question

for the polynomial, $p(x) = x^3 - 13x^2 + 72x - 160$, 5 is one of the zeros. find all remaining zeros.

the remaining zeros are  .
(simplify your answer. type your answer in the form $a + bi$. use a comma to separate answers as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Zeros" ], "new_concepts": [ "Synthetic Division", "Quadratic Formula" ], "current_concepts": [ "Polynomial Zeros", "Synthetic Division", "Quadratic Formula" ] } ## Perform synthetic division of $P(x)$ by $x - 5$ $$ \begin{array}{r|rrrr} 5 & 1 & -13 & 72 & -160 \ & & 5 & -40 & 160 \ \hline & 1 & -8 & 32 & 0 \end{array} $$ $$ Q(x) = x^2 - 8x + 32 $$ ## Solve the quadratic equation $x^2 - 8x + 32 = 0$ $$ \begin{aligned} x &= \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(32)}}{2(1)} \ x &= \frac{8 \pm \sqrt{64 - 128}}{2} \ x &= \frac{8 \pm \sqrt{-64}}{2} \ x &= \frac{8 \pm 8i}{2} \ x &= 4 \pm 4i \end{aligned} $$ The remaining zeros are $4+4i, 4-4i$. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Zeros" ] }

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Question

Explanation:

Perform synthetic division of \(P(x)\) by \(x - 5\)

Solve the quadratic equation \(x^2 - 8x + 32 = 0\)

Answer: