Question

the diagram represents \\(6x^2 - 7x + 2\\) with a factor of \\(2x - 1\\).

what is the other factor of \\(6x^2 - 7x + 2\\)?

\\(3x - 2\\)
\\(3x - 1\\)
\\(3x + 1\\)
\\(3x + 2\\)

Explanation:

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<reasoning>

Identify the relationship in the area model

The area model represents the multiplication of two binomial factors to yield the quadratic expression \(6x^2 - 7x + 2\). One factor is given along the top column headers:
\[
2x - 1
\]
The other factor is represented by the row headers containing the question marks:
\[
?x + ?
\]

Determine the first term of the missing factor

The product of the first term of the missing factor and the first term of the known factor (\(2x\)) must equal the top-left cell value (\(6x^2\)):
\[
\text{First Term} \cdot 2x = 6x^2 \implies \text{First Term} = \frac{6x^2}{2x} = 3x
\]

Determine the second term of the missing factor

The product of the second term of the missing factor and the second term of the known factor (\(-1\)) must equal the bottom-right cell value (\(2\)):
\[
\text{Second Term} \cdot (-1) = 2 \implies \text{Second Term} = \frac{2}{-1} = -2
\]
Combining these terms gives the other factor:
\[
3x - 2
\]
</reasoning>

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

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Answer:

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<reasoning>

Identify the relationship in the area model

The area model represents the multiplication of two binomial factors to yield the quadratic expression \(6x^2 - 7x + 2\). One factor is given along the top column headers:
\[
2x - 1
\]
The other factor is represented by the row headers containing the question marks:
\[
?x + ?
\]

Determine the first term of the missing factor

The product of the first term of the missing factor and the first term of the known factor (\(2x\)) must equal the top-left cell value (\(6x^2\)):
\[
\text{First Term} \cdot 2x = 6x^2 \implies \text{First Term} = \frac{6x^2}{2x} = 3x
\]

Determine the second term of the missing factor

The product of the second term of the missing factor and the second term of the known factor (\(-1\)) must equal the bottom-right cell value (\(2\)):
\[
\text{Second Term} \cdot (-1) = 2 \implies \text{Second Term} = \frac{2}{-1} = -2
\]
Combining these terms gives the other factor:
\[
3x - 2
\]
</reasoning>

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

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