Question

question

multiply: \\(\frac{n^2 - 7n}{n^2 + 2n + 1} \cdot \frac{n + 1}{2n}\\)

note: enter answer in factored form

provide your answer below:

Explanation:

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<reasoning>

Factor the first numerator

Using the Greatest Common Factor Factoring knowledge point
\[
n^2 - 7n = n(n - 7)
\]

Factor the first denominator

Using the Perfect Square Trinomials knowledge point
\[
n^2 + 2n + 1 = (n + 1)^2
\]

Rewrite the multiplication expression

Using the Multiplying Rational Expressions knowledge point
We substitute the factored forms back into the original expression:
\[
\frac{n(n - 7)}{(n + 1)^2} \cdot \frac{n + 1}{2n}
\]

Simplify the expression

Using the Simplifying Rational Expressions knowledge point
We combine the fractions and cancel the common factors \(n\) and \((n + 1)\) from the numerator and denominator:
\[
\frac{n(n - 7)(n + 1)}{2n(n + 1)^2} = \frac{n - 7}{2(n + 1)}
\]
</reasoning>

<answer>
\(\frac{n - 7}{2(n + 1)}\)
</answer>

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"question_type": "Free Response",
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"Mathematics",
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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Factor the first numerator

Using the Greatest Common Factor Factoring knowledge point
\[
n^2 - 7n = n(n - 7)
\]

Factor the first denominator

Using the Perfect Square Trinomials knowledge point
\[
n^2 + 2n + 1 = (n + 1)^2
\]

Rewrite the multiplication expression

Using the Multiplying Rational Expressions knowledge point
We substitute the factored forms back into the original expression:
\[
\frac{n(n - 7)}{(n + 1)^2} \cdot \frac{n + 1}{2n}
\]

Simplify the expression

Using the Simplifying Rational Expressions knowledge point
We combine the fractions and cancel the common factors \(n\) and \((n + 1)\) from the numerator and denominator:
\[
\frac{n(n - 7)(n + 1)}{2n(n + 1)^2} = \frac{n - 7}{2(n + 1)}
\]
</reasoning>

<answer>
\(\frac{n - 7}{2(n + 1)}\)
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Multiplying Rational Expressions"
]
}
</post_analysis>