identify the equation for the graph.

* $y = (x-1)^2(x-3)(x-5)^2$
* $y = (x-1)(x-3)^2(x-5)$
* $y = (x-1)^3(x-3)(x-5)^3$
* $y = (x-1)(x-3)^3(x-5)$

Question

identify the equation for the graph.

* $y = (x-1)^2(x-3)(x-5)^2$
* $y = (x-1)(x-3)^2(x-5)$
* $y = (x-1)^3(x-3)(x-5)^3$
* $y = (x-1)(x-3)^3(x-5)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "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" ] } ## Identify the roots from the graph Using the *Graph Behavior at Roots* knowledge point $$ \begin{aligned} &\text{Roots are located at the } x\text{-intercepts:}\ &x = 1 \quad (\text{crosses directly})\ &x = 3 \quad (\text{tangent/flattens, inflection point style})\ &x = 5 \quad (\text{crosses directly}) \end{aligned} $$ ## Determine the multiplicity of each root Using the *Polynomial Multiplicity* knowledge point $$ \begin{aligned} &\text{At } x = 1: \text{ crosses linearly } \implies \text{multiplicity } = 1\ &\text{At } x = 3: \text{ crosses with inflection (flat) } \implies \text{multiplicity } = 3\ &\text{At } x = 5: \text{ crosses linearly } \implies \text{multiplicity } = 1 \end{aligned} $$ ## Analyze the end behavior and leading coefficient Using the *Polynomial End Behavior* knowledge point $$ \begin{aligned} &\text{As } x \to \infty, y \to \infty\ &\text{As } x \to -\infty, y \to -\infty\ &\text{Degree } = 1 + 3 + 1 = 5 \quad (\text{odd})\ &\text{Leading coefficient } a > 0 \implies a = 1 \end{aligned} $$ ## Construct the polynomial equation Using the *Polynomial Multiplicity* knowledge point $$ y = (x-1)^1(x-3)^3(x-5)^1 = (x-1)(x-3)^3(x-5) $$

(A) $y = (x-1)^2(x-3)(x-5)^2$

(B) $y = (x-1)(x-3)^2(x-5)$

(C) $y = (x-1)^3(x-3)(x-5)^3$

(D) $y = (x-1)(x-3)^3(x-5)$

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

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QUESTION IMAGE

Question

Explanation:

Identify the roots from the graph

Determine the multiplicity of each root

Analyze the end behavior and leading coefficient

Construct the polynomial equation

Answer: