Question

the binomial \\((y - 2)\\) is a factor of \\(y^2 - 10y + 16\\). what is the other factor?

\\((y - 5)\\)
\\((y + 5)\\)
\\((y - 8)\\)
\\((y + 8)\\)

Explanation:

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<reasoning>

Identify the given polynomial and known factor

\[

$$\begin{aligned} P(y) &= y^2 - 10y + 16 \\ F_1(y) &= y - 2 \end{aligned}$$

\]

Find the other factor by division or factoring

\[

$$\begin{aligned} y^2 - 10y + 16 &= (y - 2)(y - k) \\ 16 &= (-2)(-k) \implies k = 8 \\ -10 &= -2 - k \implies -10 = -10 \\ F_2(y) &= y - 8 \end{aligned}$$

\]

Match with the given options

\[
\text{The other factor is } (y - 8).
\]
</reasoning>

<answer>
<mcq-option>(y - 5)</mcq-option>
<mcq-option>(y + 5)</mcq-option>
<mcq-correct>(y - 8)</mcq-correct>
<mcq-option>(y + 8)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
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"Mathematics",
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]
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Answer:

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"new_concepts": [],
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</pre_analysis>

<reasoning>

Identify the given polynomial and known factor

\[

$$\begin{aligned} P(y) &= y^2 - 10y + 16 \\ F_1(y) &= y - 2 \end{aligned}$$

\]

Find the other factor by division or factoring

\[

$$\begin{aligned} y^2 - 10y + 16 &= (y - 2)(y - k) \\ 16 &= (-2)(-k) \implies k = 8 \\ -10 &= -2 - k \implies -10 = -10 \\ F_2(y) &= y - 8 \end{aligned}$$

\]

Match with the given options

\[
\text{The other factor is } (y - 8).
\]
</reasoning>

<answer>
<mcq-option>(y - 5)</mcq-option>
<mcq-option>(y + 5)</mcq-option>
<mcq-correct>(y - 8)</mcq-correct>
<mcq-option>(y + 8)</mcq-option>
</answer>

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