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

-2
-1
1
2

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?

-2
-1
1
2

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 $$ \begin{aligned} f(x) &= x^3 - 3x - 2 \ f(-1) &= (-1)^3 - 3(-1) - 2 = -1 + 3 - 2 = 0 \ f(2) &= (2)^3 - 3(2) - 2 = 8 - 6 - 2 = 0 \end{aligned} $$ ## Factor the polynomial to determine multiplicities $$ \begin{aligned} f(x) &= (x + 1)(x^2 - x - 2) \ f(x) &= (x + 1)(x - 2)(x + 1) \ f(x) &= (x + 1)^2(x - 2) \end{aligned} $$ ## Identify the double root $$ \begin{aligned} &\text{Root } x = -1 \text{ has multiplicity 2 (double root, tangent to the x-axis).} \ &\text{Root } x = 2 \text{ has multiplicity 1 (single root, crosses the x-axis).} \end{aligned} $$

(A) $-2$

(B) $-1$

(C) $1$

(D) $2$

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QUESTION IMAGE

Question

Explanation:

Find the roots of the cubic function

Factor the polynomial to determine multiplicities

Identify the double root

Answer: