Question

practice factoring trinomials with a leading coefficient of 1 and a positive constant term.

the binomial \\((x + 5)\\) is a factor of \\(x^2 + 8x + 15\\). what is the other factor?

\\((x + 3)\\)
\\((x + 7)\\)
\\((x + 12)\\)
\\((x + 13)\\)

Explanation:

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<reasoning>

Identify the given polynomial and known factor

\[

$$\begin{aligned} P(x) &= x^2 + 8x + 15 \\ F_1(x) &= x + 5 \end{aligned}$$

\]

Find the other factor by division or factorization

\[

$$\begin{aligned} x^2 + 8x + 15 &= (x + 5)(x + a) \\ x^2 + 8x + 15 &= x^2 + (5 + a)x + 5a \\ 5a &= 15 \implies a = 3 \\ F_2(x) &= x + 3 \end{aligned}$$

\]

Match with the given options

\[
\text{The other factor is } (x + 3).
\]
</reasoning>

<answer>
<mcq-correct>(x + 3)</mcq-correct>
<mcq-option>(x + 7)</mcq-option>
<mcq-option>(x + 12)</mcq-option>
<mcq-option>(x + 13)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the given polynomial and known factor

\[

$$\begin{aligned} P(x) &= x^2 + 8x + 15 \\ F_1(x) &= x + 5 \end{aligned}$$

\]

Find the other factor by division or factorization

\[

$$\begin{aligned} x^2 + 8x + 15 &= (x + 5)(x + a) \\ x^2 + 8x + 15 &= x^2 + (5 + a)x + 5a \\ 5a &= 15 \implies a = 3 \\ F_2(x) &= x + 3 \end{aligned}$$

\]

Match with the given options

\[
\text{The other factor is } (x + 3).
\]
</reasoning>

<answer>
<mcq-correct>(x + 3)</mcq-correct>
<mcq-option>(x + 7)</mcq-option>
<mcq-option>(x + 12)</mcq-option>
<mcq-option>(x + 13)</mcq-option>
</answer>

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