Question

which terms complete the factorization of \\(x^2 + 27x + 162\\) represented by the model?

• 27, 9x, 18x
• 9, 9x, 18x
• 27, 9x, 27x
• 9, 9x, 27x

Explanation:

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<reasoning>

Analyze the area model structure

The area model represents the multiplication of two binomials to form the trinomial \(x^2 + 27x + 162\).
The row headers are on the left: \(x\) and \(18\).
The column headers are on the top: \(x\) and an unknown constant \(a\).
The cells inside the grid represent the products of the row and column headers:

• Top-left cell: \(x \cdot x = x^2\)
• Bottom-right cell: \(18 \cdot a = 162\)

Find the missing column header

Using the bottom-right cell:
\[
18 \cdot a = 162 \implies a = \frac{162}{18} = 9
\]
Thus, the missing term at the top is \(9\).

Find the remaining product cells

The two remaining cells represent the \(x\)-terms that sum to the middle term \(27x\):

• Bottom-left cell: \(18 \cdot x = 18x\)
• Top-right cell: \(x \cdot a = 9x\)

The terms that complete the factorization model are the missing header \(9\) and the two partial product terms \(9x\) and \(18x\).
</reasoning>

<answer>
<mcq-option>O 27, 9x, 18x</mcq-option>
<mcq-correct>O 9, 9x, 18x</mcq-correct>
<mcq-option>O 27, 9x, 27x</mcq-option>
<mcq-option>O 9, 9x, 27x</mcq-option>
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Analyze the area model structure

The area model represents the multiplication of two binomials to form the trinomial \(x^2 + 27x + 162\).
The row headers are on the left: \(x\) and \(18\).
The column headers are on the top: \(x\) and an unknown constant \(a\).
The cells inside the grid represent the products of the row and column headers:

• Top-left cell: \(x \cdot x = x^2\)
• Bottom-right cell: \(18 \cdot a = 162\)

Find the missing column header

Using the bottom-right cell:
\[
18 \cdot a = 162 \implies a = \frac{162}{18} = 9
\]
Thus, the missing term at the top is \(9\).

Find the remaining product cells

The two remaining cells represent the \(x\)-terms that sum to the middle term \(27x\):

• Bottom-left cell: \(18 \cdot x = 18x\)
• Top-right cell: \(x \cdot a = 9x\)

The terms that complete the factorization model are the missing header \(9\) and the two partial product terms \(9x\) and \(18x\).
</reasoning>

<answer>
<mcq-option>O 27, 9x, 18x</mcq-option>
<mcq-correct>O 9, 9x, 18x</mcq-correct>
<mcq-option>O 27, 9x, 27x</mcq-option>
<mcq-option>O 9, 9x, 27x</mcq-option>
</answer>

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