Question

determining the model of a factorization

which model shows the correct factorization of \\(x^2 - x - 2\\)?

Explanation:

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<reasoning>

Factor the quadratic trinomial

\[
x^2 - x - 2 = (x - 2)(x + 1)
\]

Determine the dimensions of the algebra tile model

The factors represent the side lengths of the rectangle:

• One side has length \(x - 2\), represented by one \(+x\) tile and two \(-1\) tiles.
• The other side has length \(x + 1\), represented by one \(+x\) tile and one \(+1\) tile.

Match with the given options

• Option 1: Left side has \(+x\) and \(-1\). Top side has \(+x\), \(+1\), \(+1\). This represents \((x - 1)(x + 2) = x^2 + x - 2\).
• Option 2: Left side has \(+x\) and \(+1\). Top side has \(+x\), \(-1\), \(-1\). This represents \((x + 1)(x - 2) = x^2 - x - 2\).
• Option 3: Left side has \(+x\) and \(-1\). Top side has \(+x\), \(+1\), \(-1\). This represents \((x - 1)(x) = x^2 - x\).
• Option 4: Left side has \(+x\) and \(+1\). Top side has \(+x\), \(+1\), \(+1\). This represents \((x + 1)(x + 2) = x^2 + 3x + 2\).

Therefore, the second model is correct.
</reasoning>

<answer>
<mcq-option>(A) First model: Left side \(x - 1\), Top side \(x + 2\)</mcq-option>
<mcq-correct>(B) Second model: Left side \(x + 1\), Top side \(x - 2\)</mcq-correct>
<mcq-option>(C) Third model: Left side \(x - 1\), Top side \(x\)</mcq-option>
<mcq-option>(D) Fourth model: Left side \(x + 1\), Top side \(x + 2\)</mcq-option>
</answer>

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"Mathematics",
"Algebra",
"Algebra Tiles"
]
}
</post_analysis>

Answer:

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<reasoning>

Factor the quadratic trinomial

\[
x^2 - x - 2 = (x - 2)(x + 1)
\]

Determine the dimensions of the algebra tile model

The factors represent the side lengths of the rectangle:

• One side has length \(x - 2\), represented by one \(+x\) tile and two \(-1\) tiles.
• The other side has length \(x + 1\), represented by one \(+x\) tile and one \(+1\) tile.

Match with the given options

• Option 1: Left side has \(+x\) and \(-1\). Top side has \(+x\), \(+1\), \(+1\). This represents \((x - 1)(x + 2) = x^2 + x - 2\).
• Option 2: Left side has \(+x\) and \(+1\). Top side has \(+x\), \(-1\), \(-1\). This represents \((x + 1)(x - 2) = x^2 - x - 2\).
• Option 3: Left side has \(+x\) and \(-1\). Top side has \(+x\), \(+1\), \(-1\). This represents \((x - 1)(x) = x^2 - x\).
• Option 4: Left side has \(+x\) and \(+1\). Top side has \(+x\), \(+1\), \(+1\). This represents \((x + 1)(x + 2) = x^2 + 3x + 2\).

Therefore, the second model is correct.
</reasoning>

<answer>
<mcq-option>(A) First model: Left side \(x - 1\), Top side \(x + 2\)</mcq-option>
<mcq-correct>(B) Second model: Left side \(x + 1\), Top side \(x - 2\)</mcq-correct>
<mcq-option>(C) Third model: Left side \(x - 1\), Top side \(x\)</mcq-option>
<mcq-option>(D) Fourth model: Left side \(x + 1\), Top side \(x + 2\)</mcq-option>
</answer>

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