completely factor the polynomial, if possible.

\x^2 + 4x + 16\

Question

completely factor the polynomial, if possible.

\x^2 + 4x + 16\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Factoring Trinomials", "Prime Polynomials" ], "new_concepts": [], "current_concepts": [ "Factoring Trinomials", "Prime Polynomials" ] } ## Identify the polynomial and factoring strategy Using the *Factoring Trinomials* knowledge point $$ x^2 + 4x + 16 $$ ## Analyze the coefficients for factoring Using the *Factoring Trinomials* knowledge point $$ \begin{aligned} &\text{Standard form: } ax^2 + bx + c \ &a = 1, \quad b = 4, \quad c = 16 \ &\text{Find two integers with product } ac = 16 \text{ and sum } b = 4 \end{aligned} $$ ## List factor pairs of 16 Using the *Factoring Trinomials* knowledge point $$ \begin{aligned} &1 \times 16 \implies 1 + 16 = 17 \ &2 \times 8 \implies 2 + 8 = 10 \ &4 \times 4 \implies 4 + 4 = 8 \end{aligned} $$ ## Determine if the polynomial is prime Using the *Prime Polynomials* knowledge point $$ \begin{aligned} &\text{No integer factor pairs of 16 sum to 4.} \ &\text{Therefore, the polynomial cannot be factored over the integers.} \end{aligned} $$ Prime Polynomial { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Prime Polynomials" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the polynomial and factoring strategy

Analyze the coefficients for factoring

List factor pairs of 16

Determine if the polynomial is prime

Answer: