which of the following could represent the graph of (f(x) = x^4 + x^3 - 8x^2 - 12x)?

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Factored Polynomials", "Polynomial Graph Behavior", "Zeroes of a Function" ], "new_concepts": [], "current_concepts": [ "Factored Polynomials", "Polynomial Graph Behavior", "Zeroes of a Function", "Root Multiplicity" ] } ## Factor the polynomial function $$ \begin{aligned} f(x) &= x^4 + x^3 - 8x^2 - 12x \ &= x(x^3 + x^2 - 8x - 12) \ &= x(x + 2)^2(x - 3) \end{aligned} $$ ## Determine the roots and multiplicities $$ \begin{aligned} x &= 0 \quad (\text{multiplicity } 1) \ x &= -2 \quad (\text{multiplicity } 2) \ x &= 3 \quad (\text{multiplicity } 1) \end{aligned} $$ ## Match roots and behavior to the graph $$ \begin{aligned} &\text{At } x = -2: \text{graph touches the x-axis and turns around (local maximum at } (0,0) \text{ is incorrect, tangent at } x = -2\text{)} \ &\text{At } x = 0: \text{graph crosses the x-axis} \ &\text{At } x = 3: \text{graph crosses the x-axis} \ &\text{The first graph shows: tangent/touching at } x = 0 \text{ (incorrect), crossing at } x = -2 \text{ and } x = 3. \ &\text{Therefore, the first graph is incorrect, and the correct graph must have a tangent point at } x = -2. \end{aligned} $$

(A) The first graph shown, which has x-intercepts at $x = -2$, $x = 0$, and $x = 3$, but touches and turns around at $x = 0$.

(B) The second graph (partially visible), which correctly touches and turns around at $x = -2$, and crosses the x-axis at $x = 0$ and $x = 3$.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Graph Behavior" ] }

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Explanation:

Factor the polynomial function

Determine the roots and multiplicities

Match roots and behavior to the graph

Answer: