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>

Set up the polynomial equation

\[
2x^2 + 13x + 6 = (x + 6)(ax + b)
\]

Solve for the coefficients

\[

$$\begin{aligned} a &= \frac{2x^2}{x} = 2x \\ b &= \frac{6}{6} = 1 \end{aligned}$$

\]

Verify the middle term

\[
(x + 6)(2x + 1) = 2x^2 + x + 12x + 6 = 2x^2 + 13x + 6
\]
</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>

Set up the polynomial equation

\[
2x^2 + 13x + 6 = (x + 6)(ax + b)
\]

Solve for the coefficients

\[

$$\begin{aligned} a &= \frac{2x^2}{x} = 2x \\ b &= \frac{6}{6} = 1 \end{aligned}$$

\]

Verify the middle term

\[
(x + 6)(2x + 1) = 2x^2 + x + 12x + 6 = 2x^2 + 13x + 6
\]
</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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