Which expression is equivalent to 3/(x^(2) + x) - 1/(x + 2), where x neq 0, x neq -1, and...
GMAT Advanced Math : (Adv_Math) Questions
Which expression is equivalent to \(\frac{3}{x^{2} + x} - \frac{1}{x + 2}\), where \(x \neq 0\), \(x \neq -1\), and \(x \neq -2\)?
1. INFER the approach needed
- When subtracting rational expressions, we need a common denominator
- But first, let's see if we can factor any denominators to make this easier
2. SIMPLIFY by factoring the first denominator
- Factor \(x^2 + x\): Look for common factors
- \(x^2 + x = x(x + 1)\)
- Now our expression is: \(\frac{3}{x(x + 1)} - \frac{1}{x + 2}\)
3. INFER the common denominator strategy
- First denominator: \(x(x + 1)\)
- Second denominator: \((x + 2)\)
- These share no common factors, so LCD = \(x(x + 1)(x + 2)\)
4. SIMPLIFY by rewriting with common denominators
\(\frac{3}{x(x + 1)} - \frac{1}{x + 2} = \frac{3(x + 2)}{x(x + 1)(x + 2)} - \frac{x(x + 1)}{x(x + 1)(x + 2)}\)
5. SIMPLIFY by subtracting the fractions
\(= \frac{3(x + 2) - x(x + 1)}{x(x + 1)(x + 2)}\)
6. SIMPLIFY the numerator through expansion
- Expand \(3(x + 2) = 3x + 6\)
- Expand \(x(x + 1) = x^2 + x\)
- So: \(3(x + 2) - x(x + 1) = 3x + 6 - x^2 - x = -x^2 + 2x + 6\)
Answer: D \(\frac{-x^2 + 2x + 6}{x(x + 1)(x + 2)}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Poor SIMPLIFY execution: Students correctly identify the need for a common denominator but make errors when expanding the numerator.
A typical mistake: \(3(x + 2) - x(x + 1) = 3x + 6 - x^2 + x\) (forgetting the negative sign distributes)
This gives numerator \(-x^2 + 4x + 6\) instead of \(-x^2 + 2x + 6\), leading to confusion and guessing among the answer choices.
Second Most Common Error:
Incomplete INFER reasoning about common denominators: Students find a common denominator but miss including all factors.
They might use \(x(x + 1)\) or \((x + 1)(x + 2)\) as the LCD instead of the full \(x(x + 1)(x + 2)\).
This may lead them to select Choice B \(\frac{-x^2 + 2x + 6}{x(x + 1)}\) or Choice C \(\frac{-x^2 + 2x + 6}{(x + 1)(x + 2)}\).
The Bottom Line:
This problem tests systematic algebraic manipulation - students must execute multiple SIMPLIFY steps flawlessly while maintaining the correct strategy throughout.