which polynomial function has a leading coefficient of 1, roots -2 and 7 with multiplicity 1, and root 5 with multiplicity 2?

$f(x) = 2(x + 7)(x + 5)(x - 2)$
$f(x) = 2(x - 7)(x - 5)(x + 2)$
$f(x) = (x + 7)(x + 5)(x + 5)(x - 2)$
$f(x) = (x - 7)(x - 5)(x - 5)(x + 2)$

Question

which polynomial function has a leading coefficient of 1, roots -2 and 7 with multiplicity 1, and root 5 with multiplicity 2?

$f(x) = 2(x + 7)(x + 5)(x - 2)$
$f(x) = 2(x - 7)(x - 5)(x + 2)$
$f(x) = (x + 7)(x + 5)(x + 5)(x - 2)$
$f(x) = (x - 7)(x - 5)(x - 5)(x + 2)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Factored Form of Polynomials", "Multiplicity of Roots" ], "new_concepts": [], "current_concepts": [ "Factored Form of Polynomials", "Multiplicity of Roots", "Roots of Polynomials" ] } ## Identify factors from given roots and multiplicities $$ \begin{aligned} &\text{Root } x = -2 \text{ with multiplicity 1} \implies (x + 2)^1\ &\text{Root } x = 7 \text{ with multiplicity 1} \implies (x - 7)^1\ &\text{Root } x = 5 \text{ with multiplicity 2} \implies (x - 5)^2 = (x - 5)(x - 5) \end{aligned} $$ ## Construct the polynomial function $$ \begin{aligned} &f(x) = a(x - 7)(x - 5)(x - 5)(x + 2)\ &\text{Given leading coefficient } a = 1 \implies f(x) = (x - 7)(x - 5)(x - 5)(x + 2) \end{aligned} $$ ## Match with the given options $$ f(x) = (x - 7)(x - 5)(x - 5)(x + 2) $$

(A) $f(x) = 2(x + 7)(x + 5)(x - 2)$

(B) $f(x) = 2(x - 7)(x - 5)(x + 2)$

(C) $f(x) = (x + 7)(x + 5)(x + 5)(x - 2)$

(D) $f(x) = (x - 7)(x - 5)(x - 5)(x + 2)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Factored Form of Polynomials" ] }

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QUESTION IMAGE

Question

Explanation:

Identify factors from given roots and multiplicities

Construct the polynomial function

Match with the given options

Answer: