Question

modeling polynomials with algebra tiles

which geometric model using algebra tiles represents the factorization of (x^2 - 5x + 6)?

Explanation:

<pre_analysis>
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"skills_matched": [
"step_cot",
"multiple_choice_formatter",
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"tutor"
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"concepts_used": [
"Algebra Tiles",
"Factoring Trinomials"
],
"new_concepts": [],
"current_concepts": [
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</pre_analysis>

<reasoning>

Factor the quadratic trinomial

Using the Factoring Trinomials knowledge point, we find two integers that multiply to \(+6\) and add up to \(-5\).
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]

Determine the dimensions of the algebra tile model

Using the Algebra Tiles knowledge point, the side lengths of the rectangle represent the factors.
\[
\text{Width} = x - 3,\quad \text{Height} = x - 2
\]

Match the factors to the visual models

We examine the headers along the top and left edges of each option:

• Option 1: Top header is \(x - 3\) (one \(+x\) tile and three \(-1\) tiles). Left header is \(x - 2\) (one \(+x\) tile and two \(-1\) tiles).
• Option 2: Top header is \(x - 2\). Left header is \(x - 2\).
• Option 3: Top header is \(x - 2\). Left header is \(x - 3\).
• Option 4: Top header is \(x - 3\). Left header is \(x - 3\).

Identify the correct geometric representation

Option 1 correctly represents the product of \((x - 3)\) along the top and \((x - 2)\) along the left:

• The top header has one \(+x\) tile and three \(-1\) tiles: \(x - 3\).
• The left header has one \(+x\) tile and two \(-1\) tiles: \(x - 2\).
• The inner area contains one \(x^2\) tile, five \(-x\) tiles, and six \(+1\) tiles, representing \(x^2 - 5x + 6\).

</reasoning>

<answer>
<mcq-correct>(A) Top header: \(x - 3\), Left header: \(x - 2\)</mcq-correct>
<mcq-option>(B) Top header: \(x - 2\), Left header: \(x - 2\)</mcq-option>
<mcq-option>(C) Top header: \(x - 2\), Left header: \(x - 3\)</mcq-option>
<mcq-option>(D) Top header: \(x - 3\), Left header: \(x - 3\)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Algebra Tiles"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
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"concepts_used": [
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"Factoring Trinomials"
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"new_concepts": [],
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</pre_analysis>

<reasoning>

Factor the quadratic trinomial

Using the Factoring Trinomials knowledge point, we find two integers that multiply to \(+6\) and add up to \(-5\).
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]

Determine the dimensions of the algebra tile model

Using the Algebra Tiles knowledge point, the side lengths of the rectangle represent the factors.
\[
\text{Width} = x - 3,\quad \text{Height} = x - 2
\]

Match the factors to the visual models

We examine the headers along the top and left edges of each option:

• Option 1: Top header is \(x - 3\) (one \(+x\) tile and three \(-1\) tiles). Left header is \(x - 2\) (one \(+x\) tile and two \(-1\) tiles).
• Option 2: Top header is \(x - 2\). Left header is \(x - 2\).
• Option 3: Top header is \(x - 2\). Left header is \(x - 3\).
• Option 4: Top header is \(x - 3\). Left header is \(x - 3\).

Identify the correct geometric representation

Option 1 correctly represents the product of \((x - 3)\) along the top and \((x - 2)\) along the left:

• The top header has one \(+x\) tile and three \(-1\) tiles: \(x - 3\).
• The left header has one \(+x\) tile and two \(-1\) tiles: \(x - 2\).
• The inner area contains one \(x^2\) tile, five \(-x\) tiles, and six \(+1\) tiles, representing \(x^2 - 5x + 6\).

</reasoning>

<answer>
<mcq-correct>(A) Top header: \(x - 3\), Left header: \(x - 2\)</mcq-correct>
<mcq-option>(B) Top header: \(x - 2\), Left header: \(x - 2\)</mcq-option>
<mcq-option>(C) Top header: \(x - 2\), Left header: \(x - 3\)</mcq-option>
<mcq-option>(D) Top header: \(x - 3\), Left header: \(x - 3\)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Algebra Tiles"
]
}
</post_analysis>