graph the function $f(x) = x^3 - 3x - 2$. based on the graph, which value for $x$ is a double root of this function?

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Question

graph the function $f(x) = x^3 - 3x - 2$. based on the graph, which value for $x$ is a double root of this function?

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Polynomial Roots", "Cubic Function Properties" ], "new_concepts": [], "current_concepts": [ "Polynomial Roots", "Cubic Function Properties", "Graphical Solutions" ] } ## Find the roots of the cubic function $$ \begin{aligned} f(x) &= x^3 - 3x - 2 \ &= x^3 + x^2 - x^2 - x - 2x - 2 \ &= x^2(x + 1) - x(x + 1) - 2(x + 1) \ &= (x + 1)(x^2 - x - 2) \end{aligned} $$ ## Factor the remaining quadratic expression $$ \begin{aligned} x^2 - x - 2 &= (x - 2)(x + 1) \ f(x) &= (x + 1)^2(x - 2) \end{aligned} $$ ## Identify the double root $$ \begin{aligned} &\text{Roots: } x = -1 \text{ (multiplicity 2), } x = 2 \text{ (multiplicity 1)} \ &\text{Double root: } x = -1 \end{aligned} $$

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QUESTION IMAGE

Question

Explanation:

Find the roots of the cubic function

Factor the remaining quadratic expression

Identify the double root

Answer: