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\\)

Explanation:

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<reasoning>

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
\[

$$\begin{aligned} &(x + 6)(2x + 1) = 2x^2 + x + 12x + 6 = 2x^2 + 13x + 6 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x+3\)</mcq-option>
<mcq-option>(B) \(x+6\)</mcq-option>
<mcq-correct>(C) \(2x+1\)</mcq-correct>
<mcq-option>(D) \(2x+3\)</mcq-option>
</answer>

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Answer:

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<reasoning>

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
\[

$$\begin{aligned} &(x + 6)(2x + 1) = 2x^2 + x + 12x + 6 = 2x^2 + 13x + 6 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x+3\)</mcq-option>
<mcq-option>(B) \(x+6\)</mcq-option>
<mcq-correct>(C) \(2x+1\)</mcq-correct>
<mcq-option>(D) \(2x+3\)</mcq-option>
</answer>

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