a)

$ 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

a)

$ 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": [], "new_concepts": [ "Polynomial Graphs", "Multiplicity of Roots" ], "current_concepts": [ "Polynomial Factors", "Polynomial Graphs", "Multiplicity of Roots" ] } ## Identify the x-intercepts and their behavior $$ \begin{aligned} &\text{Intercept at } x = 1 \text{ (tangent to the x-axis, multiplicity 2)} \ &\text{Intercept at } x = 4 \text{ (crosses the x-axis, multiplicity 1)} \end{aligned} $$ ## Formulate the polynomial equation $$ \begin{aligned} &f(x) = a(x - 1)^2(x - 4) \end{aligned} $$ ## Determine the leading coefficient using the y-intercept $$ \begin{aligned} &f(0) = a(0 - 1)^2(0 - 4) = -4a \ &\text{From the graph, } f(0) = -4 \implies -4a = -4 \implies a = 1 \ &f(x) = (x - 1)^2(x - 4) \end{aligned} $$

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 Factors" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the x-intercepts and their behavior

Formulate the polynomial equation

Determine the leading coefficient using the y-intercept

Answer: