solve the equation $x^3 - 5x^2 + 2x + 8 = 0$ given that $-1$ is a zero of $f(x) = x^3 - 5x^2 + 2x + 8$.

the solution set is $\{\quad\}$. (use a comma to separate answers as needed.)

Question

solve the equation $x^3 - 5x^2 + 2x + 8 = 0$ given that $-1$ is a zero of $f(x) = x^3 - 5x^2 + 2x + 8$.

the solution set is $\{\quad\}$. (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", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Synthetic Division", "Polynomial Zeros" ], "new_concepts": [ "Solving Cubic Equations" ], "current_concepts": [ "Synthetic Division", "Polynomial Zeros", "Solving Cubic Equations" ] } ## Divide the polynomial using synthetic division Using the *Synthetic Division* knowledge point $$ \begin{array}{r|rrrr} -1 & 1 & -5 & 2 & 8 \ & & -1 & 6 & -8 \ \hline & 1 & -6 & 8 & 0 \end{array} $$ ## Factor the remaining quadratic equation Using the *Polynomial Zeros* knowledge point $$ \begin{aligned} &x^2 - 6x + 8 = 0 \ &(x - 2)(x - 4) = 0 \ &x = 2 \quad \text{or} \quad x = 4 \end{aligned} $$ ## State the complete solution set Using the *Polynomial Zeros* knowledge point $$ x \in \{-1, 2, 4\} $$ Solve the equation $x^3 - 5x^2 + 2x + 8 = 0$ given that $-1$ is a zero of $f(x) = x^3 - 5x^2 + 2x + 8$. The solution set is {$-1, 2, 4$}. (Use a comma to separate answers as needed.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Solving Cubic Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Divide the polynomial using synthetic division

Factor the remaining quadratic equation

State the complete solution set

Answer: