graph the polynomial function.

$f(x) = (x + 2)(x + 3)(x + 5)^2$

a.
b.
c.
d.

Question

graph the polynomial function.

$f(x) = (x + 2)(x + 3)(x + 5)^2$

a.
b.
c.
d.

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": [ "Finding Zeros of Factored Polynomials", "Multiplicity of Zeros", "Graph Behavior at Zeros", "End Behavior of Polynomials" ], "new_concepts": [], "current_concepts": [ "Finding Zeros of Factored Polynomials", "Multiplicity of Zeros", "Graph Behavior at Zeros", "End Behavior of Polynomials", "Y-Intercept of Polynomial" ] } ## Find the zeros and their multiplicities Using the *Finding Zeros of Factored Polynomials* and *Multiplicity of Zeros* knowledge points $$ \begin{aligned} &f(x) = (x+2)(x+3)(x+5)^2 \ &\text{Zeros: } x = -2 \text{ (multiplicity 1)}, \quad x = -3 \text{ (multiplicity 1)}, \quad x = -5 \text{ (multiplicity 2)} \end{aligned} $$ ## Determine graph behavior at each zero Using the *Graph Behavior at Zeros* knowledge point $$ \begin{aligned} &\text{At } x = -2 \text{ and } x = -3 \text{ (odd multiplicity): the graph crosses the } x\text{-axis.} \ &\text{At } x = -5 \text{ (even multiplicity): the graph is tangent to (touches and turns around at) the } x\text{-axis.} \end{aligned} $$ ## Determine the end behavior Using the *End Behavior of Polynomials* knowledge point $$ \begin{aligned} &\text{Degree } n = 1 + 1 + 2 = 4 \quad (\text{even}) \ &\text{Leading coefficient } a_n = 1 > 0 \ &\text{As } x \to \pm\infty, \quad f(x) \to \infty \end{aligned} $$ ## Calculate the y-intercept Using the *Y-Intercept of Polynomial* knowledge point $$ \begin{aligned} &f(0) = (0+2)(0+3)(0+5)^2 = 2 \cdot 3 \cdot 25 = 150 \end{aligned} $$ ## Match with the correct graph option Using the *Graph Behavior at Zeros* and *End Behavior of Polynomials* knowledge points $$ \begin{aligned} &\text{Option A: Tangent at } x = -5, \text{ crosses at } x = -3 \text{ and } x = -2. \text{ Both ends go up.} \ &\text{Option B: Crosses at } x = -5 \text{ (incorrect).} \ &\text{Option C: Crosses at } x = -5 \text{ (incorrect).} \ &\text{Option D: Tangent at } x = -5, \text{ but does not cross correctly at } -3 \text{ and } -2 \text{ (incorrect shape).} \end{aligned} $$

(A) Tangent at $x = -5$, crosses at $x = -3$ and $x = -2$, with both ends pointing upwards.

(B) Crosses at $x = -5$

(C) Crosses at $x = -5$

(D) Incorrect behavior at the zeros

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Graph Behavior at Zeros" ] }

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QUESTION IMAGE

Question

Explanation:

Find the zeros and their multiplicities

Determine graph behavior at each zero

Determine the end behavior

Calculate the y-intercept

Match with the correct graph option

Answer: