find the zeros of the polynomial function, and state the multiplicity of each.

(f(x) = x^3(x-2)^2(x+8))

the smallest zero is with multiplicity
the middle zero is with multiplicity
the largest zero is with multiplicity

Question

find the zeros of the polynomial function, and state the multiplicity of each.

(f(x) = x^3(x-2)^2(x+8))

the smallest zero is   with multiplicity
the middle zero is   with multiplicity
the largest zero is   with multiplicity

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Zeros", "Multiplicity of Zeros" ], "new_concepts": [], "current_concepts": [ "Polynomial Zeros", "Multiplicity of Zeros" ] } ## Identify the zeros of each factor $$ \begin{aligned} x^3 = 0 &\implies x = 0 \ (x-2)^2 = 0 &\implies x = 2 \ x+8 = 0 &\implies x = -8 \end{aligned} $$ ## Determine the multiplicity of each zero $$ \begin{aligned} x = 0 &\implies \text{multiplicity } 3 \ x = 2 &\implies \text{multiplicity } 2 \ x = -8 &\implies \text{multiplicity } 1 \end{aligned} $$ ## Order the zeros from smallest to largest $$ \begin{aligned} \text{Smallest zero: } & -8 \quad (\text{multiplicity } 1) \ \text{Middle zero: } & 0 \quad (\text{multiplicity } 3) \ \text{Largest zero: } & 2 \quad (\text{multiplicity } 2) \end{aligned} $$ Find the zeros of the polynomial function, and state the multiplicity of each. $f(x)=x^3(x-2)^2(x+8)$ The smallest zero is -8 with multiplicity 1 The middle zero is 0 with multiplicity 3 The largest zero is 2 with multiplicity 2 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Zeros" ] }

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Question

Explanation:

Identify the zeros of each factor

Determine the multiplicity of each zero

Order the zeros from smallest to largest

Answer: