b)

$ f(x) = (x - 1)(x - 4)^2 $

$ f(x) = (x + 1)^2(x - 4) $

$ f(x) = (x - 1)(x + 4)^2 $

$ f(x) = (x - 1)^2(x - 4) $

Question

b)

$ f(x) = (x - 1)(x - 4)^2 $

$ f(x) = (x + 1)^2(x - 4) $

$ f(x) = (x - 1)(x + 4)^2 $

$ f(x) = (x - 1)^2(x - 4) $

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Factors", "Multiplicity of Roots", "Polynomial Graphs" ], "new_concepts": [], "current_concepts": [ "Polynomial Factors", "Multiplicity of Roots", "Polynomial Graphs" ] } ## Identify roots and their multiplicities from the graph $$ \begin{aligned} &x = -4 \quad (\text{tangent to the } x\text{-axis, multiplicity } 2)\ &x = 1 \quad (\text{crosses the } x\text{-axis, multiplicity } 1) \end{aligned} $$ ## Construct the polynomial function $$ f(x) = a(x - 1)(x + 4)^2 $$ ## Determine the leading coefficient and match with options $$ \begin{aligned} &f(0) = a(0 - 1)(0 + 4)^2 = -16a\ &\text{Since } f(0) \approx -16 \text{ is negative, } a \approx 1 > 0\ &f(x) = (x - 1)(x + 4)^2 \end{aligned} $$

O $f(x) = (x-1)(x-4)^2$

O $f(x) = (x+1)^2(x-4)$

O $f(x) = (x-1)(x+4)^2$

O $f(x) = (x-1)^2(x-4)$

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

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QUESTION IMAGE

Question

Explanation:

Identify roots and their multiplicities from the graph

Construct the polynomial function

Determine the leading coefficient and match with options

Answer: