the trinomial $2x^2 + 13x + 6$ has a linear factor of $x + 6$.
$2x^2 + 13x + 6 = (x + 6)(?)$
what is the other linear factor?

$x + 3$
$x + 6$
$2x + 1$
$2x + 3$

Question

the trinomial $2x^2 + 13x + 6$ has a linear factor of $x + 6$.
$2x^2 + 13x + 6 = (x + 6)(?)$
what is the other linear factor?

$x + 3$
$x + 6$
$2x + 1$
$2x + 3$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Factoring Polynomials", "Trinomial Coefficients" ], "new_concepts": [], "current_concepts": [ "Factoring Polynomials", "Trinomial Coefficients", "Algebra Tiles" ] } ## Identify the given polynomial and known factor Using the *Factoring Polynomials* knowledge point $$ \begin{aligned} &P(x) = 2x^2 + 13x + 6\ &\text{Known factor: } (x + 6) \end{aligned} $$ ## Set up the factorization equation Using the *Trinomial Coefficients* knowledge point $$ \begin{aligned} &2x^2 + 13x + 6 = (x + 6)(ax + b) \end{aligned} $$ ## Determine the coefficients of the unknown factor Using the *Factoring Polynomials* and *Trinomial Coefficients* knowledge points $$ \begin{aligned} &a = 2 \quad (\text{since } x \cdot ax = 2x^2)\ &b = 1 \quad (\text{since } 6 \cdot b = 6) \end{aligned} $$ ## Verify the middle term Using the *Factoring Polynomials* knowledge point $$ \begin{aligned} &(x + 6)(2x + 1) = 2x^2 + x + 12x + 6 = 2x^2 + 13x + 6 \end{aligned} $$

(A) $x+3$

(B) $x+6$

(C) $2x+1$

(D) $2x+3$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Factoring Polynomials" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given polynomial and known factor

Set up the factorization equation

Determine the coefficients of the unknown factor

Verify the middle term

Answer: