c)

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

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

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

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

Question

c)

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

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

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

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

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 the roots and their multiplicities from the graph $$ \begin{aligned} &\text{Root at } x = 1 \implies \text{crosses the x-axis (multiplicity 1)} \implies \text{factor is } (x - 1)\ &\text{Root at } x = 4 \implies \text{touches and turns around (multiplicity 2)} \implies \text{factor is } (x - 4)^2 \end{aligned} $$ ## Formulate the polynomial function $$ f(x) = a(x - 1)(x - 4)^2 $$ ## Determine the leading coefficient using the y-intercept $$ \begin{aligned} &f(0) = -16 \implies a(0 - 1)(0 - 4)^2 = -16 \implies -16a = -16 \implies a = 1\ &f(x) = (x - 1)(x - 4)^2 \end{aligned} $$

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

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

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

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

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

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QUESTION IMAGE

Question

Explanation:

Identify the roots and their multiplicities from the graph

Formulate the polynomial function

Determine the leading coefficient using the y-intercept

Answer: