Question:3a^2/(a - 2b) + (2ab + 8b^2)/(2b - a)Which of the following is equivalent to the expression above for all...

1. INFER the relationship between denominators

• Given expression: \(\frac{3\mathrm{a}^2}{\mathrm{a} - 2\mathrm{b}} + \frac{2\mathrm{ab} + 8\mathrm{b}^2}{2\mathrm{b} - \mathrm{a}}\)
• Key insight: The denominators \(\mathrm{a} - 2\mathrm{b}\) and \(2\mathrm{b} - \mathrm{a}\) are opposites of each other
• Since \(2\mathrm{b} - \mathrm{a} = -(\mathrm{a} - 2\mathrm{b})\), we can rewrite with a common denominator

2. SIMPLIFY to get common denominators

• Rewrite the second fraction: \(\frac{2\mathrm{ab} + 8\mathrm{b}^2}{2\mathrm{b} - \mathrm{a}} = \frac{2\mathrm{ab} + 8\mathrm{b}^2}{-(\mathrm{a} - 2\mathrm{b})} = -\frac{2\mathrm{ab} + 8\mathrm{b}^2}{\mathrm{a} - 2\mathrm{b}}\)
• The expression becomes: \(\frac{3\mathrm{a}^2}{\mathrm{a} - 2\mathrm{b}} - \frac{2\mathrm{ab} + 8\mathrm{b}^2}{\mathrm{a} - 2\mathrm{b}}\)

3. SIMPLIFY by combining numerators

• Combine over the common denominator: \(\frac{3\mathrm{a}^2 - (2\mathrm{ab} + 8\mathrm{b}^2)}{\mathrm{a} - 2\mathrm{b}} = \frac{3\mathrm{a}^2 - 2\mathrm{ab} - 8\mathrm{b}^2}{\mathrm{a} - 2\mathrm{b}}\)

4. SIMPLIFY by factoring the numerator

• Factor \(3\mathrm{a}^2 - 2\mathrm{ab} - 8\mathrm{b}^2\) using the grouping method
• Find two numbers that multiply to \((3)(-8) = -24\) and add to \(-2\): these are \(-6\) and \(4\)
• Rewrite: \(3\mathrm{a}^2 - 6\mathrm{ab} + 4\mathrm{ab} - 8\mathrm{b}^2\)
• Factor by grouping: \(3\mathrm{a}(\mathrm{a} - 2\mathrm{b}) + 4\mathrm{b}(\mathrm{a} - 2\mathrm{b}) = (3\mathrm{a} + 4\mathrm{b})(\mathrm{a} - 2\mathrm{b})\)

5. SIMPLIFY by canceling common factors

• \(\frac{(3\mathrm{a} + 4\mathrm{b})(\mathrm{a} - 2\mathrm{b})}{\mathrm{a} - 2\mathrm{b}} = 3\mathrm{a} + 4\mathrm{b}\)

Answer: D. 3a + 4b

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(2\mathrm{b} - \mathrm{a}\) and \(\mathrm{a} - 2\mathrm{b}\) are opposites, so they attempt to find a common denominator by multiplying the denominators together, creating \((\mathrm{a} - 2\mathrm{b})(2\mathrm{b} - \mathrm{a})\) as the common denominator. This leads to an unnecessarily complex expression that becomes very difficult to simplify, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize the opposite relationship but make sign errors when converting \(\frac{2\mathrm{ab} + 8\mathrm{b}^2}{2\mathrm{b} - \mathrm{a}}\) to \(-\frac{2\mathrm{ab} + 8\mathrm{b}^2}{\mathrm{a} - 2\mathrm{b}}\). They might incorrectly write it as \(\frac{-(2\mathrm{ab} + 8\mathrm{b}^2)}{\mathrm{a} - 2\mathrm{b}} = \frac{-2\mathrm{ab} - 8\mathrm{b}^2}{\mathrm{a} - 2\mathrm{b}}\) but then forget to distribute the negative properly when combining numerators. This leads to an incorrect numerator that doesn't factor cleanly, causing confusion and often leading them to select Choice A (3a - 4b) or Choice B (3a - 2b).

The Bottom Line:

This problem tests whether students can recognize structural relationships between algebraic expressions (opposites) and execute multi-step rational expression simplification without making sign errors.