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

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

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

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

Question

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

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

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

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 the x-intercepts and their multiplicities from the graph $$ \begin{aligned} &\text{Intercept at } x = -1 \text{ touches the x-axis and turns around (even multiplicity: 2)}\ &\text{Intercept at } x = 4 \text{ crosses the x-axis (odd multiplicity: 1)} \end{aligned} $$ ## Formulate the polynomial function $$ f(x) = a(x + 1)^2(x - 4) $$ ## Determine the leading coefficient and match with the options $$ \begin{aligned} &\text{Since the end behavior is } y \to \infty \text{ as } x \to \infty \text{, } a > 0 \text{ (assume } a = 1\text{)}\ &f(x) = (x + 1)^2(x - 4) \end{aligned} $$

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

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

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

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

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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify the x-intercepts and their multiplicities from the graph

Formulate the polynomial function

Determine the leading coefficient and match with the options

Answer: