Question

what is the quotient?

\\(\frac{t + 3}{t + 4} \div (t^2 + 7t + 12)\\)

• \\((t + 3)^2\\)
• \\((t + 4)^2\\)
• \\(\frac{1}{(t + 4)^2}\\)
• \\(\frac{1}{(t + 3)^2}\\)

Explanation:

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<reasoning>

Factor the quadratic expression

\[
t^2 + 7t + 12 = (t + 3)(t + 4)
\]

Rewrite the division as multiplication by the reciprocal

\[
\frac{t + 3}{t + 4} \div (t^2 + 7t + 12) = \frac{t + 3}{t + 4} \cdot \frac{1}{(t + 3)(t + 4)}
\]

Simplify the expression

\[
\frac{t + 3}{(t + 4)(t + 3)(t + 4)} = \frac{1}{(t + 4)^2} \quad (t
eq -3)
\]
</reasoning>

<answer>
<mcq-option>(A) \((t + 3)^2\)</mcq-option>
<mcq-option>(B) \((t + 4)^2\)</mcq-option>
<mcq-correct>(C) \(\frac{1}{(t + 4)^2}\)</mcq-correct>
<mcq-option>(D) \(\frac{1}{(t + 3)^2}\)</mcq-option>
</answer>

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

Answer:

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</pre_analysis>

<reasoning>

Factor the quadratic expression

\[
t^2 + 7t + 12 = (t + 3)(t + 4)
\]

Rewrite the division as multiplication by the reciprocal

\[
\frac{t + 3}{t + 4} \div (t^2 + 7t + 12) = \frac{t + 3}{t + 4} \cdot \frac{1}{(t + 3)(t + 4)}
\]

Simplify the expression

\[
\frac{t + 3}{(t + 4)(t + 3)(t + 4)} = \frac{1}{(t + 4)^2} \quad (t
eq -3)
\]
</reasoning>

<answer>
<mcq-option>(A) \((t + 3)^2\)</mcq-option>
<mcq-option>(B) \((t + 4)^2\)</mcq-option>
<mcq-correct>(C) \(\frac{1}{(t + 4)^2}\)</mcq-correct>
<mcq-option>(D) \(\frac{1}{(t + 3)^2}\)</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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"Rational Division"
]
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