completely factor the polynomial, if possible.

\x^2 + 14xy + 16y^2\

Question

completely factor the polynomial, if possible.

\x^2 + 14xy + 16y^2\

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 structure Using the *Factoring Trinomials* knowledge point $$ x^2 + 14xy + 16y^2 $$ ## Check for perfect square trinomial form Using the *Factoring Trinomials* knowledge point $$ \begin{aligned} &\text{First term: } x^2 = (x)^2 \ &\text{Last term: } 16y^2 = (4y)^2 \ &\text{Required middle term: } 2 \cdot (x) \cdot (4y) = 8xy e 14xy \end{aligned} $$ ## Find factors for general trinomial factoring Using the *Factoring Trinomials* knowledge point $$ \begin{aligned} &\text{Find two numbers with product } ac = 1 \cdot 16 = 16 \ &\text{and sum } b = 14. \ &\text{Factors of 16: } (1, 16) \Rightarrow \text{sum } 17 \ &\phantom{\text{Factors of 16: }} (2, 8) \Rightarrow \text{sum } 10 \ &\phantom{\text{Factors of 16: }} (4, 4) \Rightarrow \text{sum } 8 \end{aligned} $$ ## Determine factorability Using the *Prime Polynomials* knowledge point $$ \text{No integer factors of 16 sum to 14.} $$ Prime { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Prime Polynomials" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the polynomial structure

Check for perfect square trinomial form

Find factors for general trinomial factoring

Determine factorability

Answer: