Which expression is equivalent to \(\frac{\mathrm{y+12}}{\mathrm{x-8}} + \frac{\mathrm{y(x-8)}}{\mathrm{x^2y-8xy}}\)?

1. INFER the strategic approach

• Looking at: \(\frac{y+12}{x-8} + \frac{y(x-8)}{x^2y-8xy}\)
• Key insight: The denominators are different, so I need a common denominator to add these fractions
• Strategy: Factor the second denominator first to see the relationship

2. SIMPLIFY by factoring the second denominator

• \(x^2y - 8xy = xy(x - 8)\)
• The expression becomes: \(\frac{y+12}{x-8} + \frac{y(x-8)}{xy(x-8)}\)
• Now I can see both denominators involve \((x-8)\)

3. INFER the common denominator needed

• First fraction has denominator: \((x-8)\)
• Second fraction has denominator: \(xy(x-8)\)
• Common denominator: \(xy(x-8)\)

4. SIMPLIFY by rewriting with common denominators

• Multiply first fraction by \(\frac{xy}{xy}\): \(\frac{xy(y+12)}{xy(x-8)}\)
• Second fraction already has the common denominator: \(\frac{y(x-8)}{xy(x-8)}\)
• Combined: \(\frac{xy(y+12) + y(x-8)}{xy(x-8)}\)

5. SIMPLIFY the numerator using distributive property

• Expand \(xy(y+12)\): \(xy^2 + 12xy\)
• Expand \(y(x-8)\): \(xy - 8y\)
• Numerator becomes: \(xy^2 + 12xy + xy - 8y\)

6. SIMPLIFY by combining like terms

• \(xy^2 + 12xy + xy - 8y = xy^2 + 13xy - 8y\)
• Final expression: \(\frac{xy^2 + 13xy - 8y}{xy(x-8)}\)

7. SIMPLIFY the denominator to match answer choices

• \(xy(x-8) = x^2y - 8xy\)
• Final answer: \(\frac{xy^2 + 13xy - 8y}{x^2y - 8xy}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make calculation errors when expanding or combining terms in the numerator.

For example, they might incorrectly expand \(xy(y+12)\) as \(xy^2 + 12y\) (forgetting the x in \(12xy\)), or make sign errors when combining \(xy - 8y\) with other terms. These algebraic mistakes in the multi-step process lead to numerators that don't match any of the given choices, causing confusion and guessing.

Second Most Common Error:

Inadequate INFER reasoning: Students attempt to add the fractions without recognizing the need to factor the second denominator first.

They might try to use \((x-8)(x^2y-8xy)\) as a common denominator instead of factoring \(x^2y-8xy = xy(x-8)\) first. This leads to much more complex expressions that are difficult to simplify and don't lead toward the given answer choices. This causes them to get stuck and abandon systematic solution.

The Bottom Line:

This problem tests the complete rational expression addition process - students must factor strategically, identify common denominators, and execute multiple algebraic steps without calculation errors. Success requires both strategic thinking and careful algebraic manipulation.