a polynomial function has a root of $-3$ with multiplicity 2, a root of 0 with multiplicity 1, a root of 1 with multiplicity 1, and a root of 3 with multiplicity 2. if the function has a positive leading coefficient and is of even degree, which could be the graph of the function?

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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": [ "Polynomial Multiplicity", "Graph Behavior at Roots", "Polynomial End Behavior" ], "new_concepts": [], "current_concepts": [ "Polynomial Multiplicity", "Graph Behavior at Roots", "Polynomial End Behavior" ] } ## Analyze the roots and their multiplicities $$ \begin{aligned} &\text{Root } x = -3 \text{ has multiplicity } 2 \implies \text{Graph touches and turns around at } x = -3.\ &\text{Root } x = 0 \text{ has multiplicity } 1 \implies \text{Graph crosses the } x\text{-axis at } x = 0.\ &\text{Root } x = 1 \text{ has multiplicity } 1 \implies \text{Graph crosses the } x\text{-axis at } x = 1.\ &\text{Root } x = 3 \text{ has multiplicity } 2 \implies \text{Graph touches and turns around at } x = 3. \end{aligned} $$ ## Determine end behavior $$ \begin{aligned} &\text{Degree} = 2 + 1 + 1 + 2 = 6 \quad (\text{Even degree})\ &\text{Leading coefficient} > 0 \implies \lim_{x \to \pm\infty} f(x) = \infty \quad (\text{Both ends point upwards}) \end{aligned} $$ ## Match with the given graph $$ \begin{aligned} &\text{At } x = -3: \text{The graph touches the } x\text{-axis at } (-3, 0) \text{ and turns back up.}\ &\text{At } x = 0: \text{The graph crosses the } x\text{-axis at } (0, 0) \text{ going downwards.}\ &\text{At } x = 1: \text{The graph crosses the } x\text{-axis at } (1, 0) \text{ going upwards.}\ &\text{At } x = 3: \text{The graph touches the } x\text{-axis at } (3, 0) \text{ and turns back up.}\ &\text{Both ends of the graph point upwards as } x \to \pm\infty. \end{aligned} $$ The visible graph perfectly matches all of these characteristics.

(A) The graph shown, which touches the x-axis at x = -3 and x = 3, crosses the x-axis at x = 0 and x = 1, and has both ends pointing upwards.

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

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Explanation:

Analyze the roots and their multiplicities

Determine end behavior

Match with the given graph

Answer: