the zeroes of the function are -2, 0, and 4

of the zeroes have a multiplicity of 2.

none
one
two
three

Question

the zeroes of the function are -2, 0, and 4

of the zeroes have a multiplicity of 2.

none
one
two
three

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Root Multiplicity", "Polynomial Graph Behavior", "Zeroes of a Function" ], "new_concepts": [], "current_concepts": [ "Zeroes of a Function", "Root Multiplicity", "Polynomial Graph Behavior" ] } ## Identify the behavior of the graph at each zero $$ \begin{aligned} &\text{At } x = -2: \text{The graph crosses the } x\text{-axis directly (linear behavior, multiplicity } = 1\text{).} \ &\text{At } x = 0: \text{The graph crosses the } x\text{-axis directly (linear behavior, multiplicity } = 1\text{).} \ &\text{At } x = 4: \text{The graph crosses the } x\text{-axis directly (linear behavior, multiplicity } = 1\text{).} \end{aligned} $$ ## Determine the number of zeroes with multiplicity of 2 $$ \begin{aligned} &\text{A multiplicity of 2 corresponds to a tangent point (touch and turn around) on the } x\text{-axis.} \ &\text{Since the graph crosses the axis at all three zeroes, none of them have a multiplicity of 2.} \ &\text{Count} = 0 \implies \text{None} \end{aligned} $$

None

One

Two

Three

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Root Multiplicity" ] }

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Question

Explanation:

Identify the behavior of the graph at each zero

Determine the number of zeroes with multiplicity of 2

Answer: