determine the multiplicity of the roots of the function $k(x) = x(x + 2)^3(x + 4)^2(x - 5)^4$.

0 has multiplicity 1
-2 has multiplicity 3
-4 has multiplicity 2
5 has multiplicity 4

at which roots does the graph cross the x-axis?
0
-2
-4
5

Question

determine the multiplicity of the roots of the function $k(x) = x(x + 2)^3(x + 4)^2(x - 5)^4$.

0 has multiplicity 1
-2 has multiplicity 3
-4 has multiplicity 2
5 has multiplicity 4

at which roots does the graph cross the x-axis?
  0
  -2
  -4
  5

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Root Multiplicity", "Factored Polynomials" ], "new_concepts": [ "Polynomial Graph Behavior" ], "current_concepts": [ "Root Multiplicity", "Factored Polynomials", "Polynomial Graph Behavior" ] } ## Identify roots and multiplicities $$ \begin{aligned} &k(x) = x^1(x+2)^3(x+4)^2(x-5)^4\ &\text{Root } x = 0 \implies \text{multiplicity } 1\ &\text{Root } x = -2 \implies \text{multiplicity } 3\ &\text{Root } x = -4 \implies \text{multiplicity } 2\ &\text{Root } x = 5 \implies \text{multiplicity } 4 \end{aligned} $$ ## Apply crossing behavior rule $$ \begin{aligned} &\text{Odd multiplicity} \implies \text{graph crosses the } x\text{-axis}\ &\text{Even multiplicity} \implies \text{graph is tangent to (touches and turns around) the } x\text{-axis} \end{aligned} $$ ## Determine crossing roots $$ \begin{aligned} &\text{Multiplicity of } 0 \text{ is } 1 \text{ (odd)} \implies \text{crosses}\ &\text{Multiplicity of } -2 \text{ is } 3 \text{ (odd)} \implies \text{crosses}\ &\text{Multiplicity of } -4 \text{ is } 2 \text{ (even)} \implies \text{does not cross}\ &\text{Multiplicity of } 5 \text{ is } 4 \text{ (even)} \implies \text{does not cross} \end{aligned} $$

(A) 0

(B) -2

(C) -4

(D) 5

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

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QUESTION IMAGE

Question

Explanation:

Identify roots and multiplicities

Apply crossing behavior rule

Determine crossing roots

Answer: