Question:Which expression is equivalent to (x^2 - 5x - 14)/(x + 2)?x - 7x - 9x + 7x + 9

1. INFER the solution strategy

• Given: \(\frac{\mathrm{x}^2 - 5\mathrm{x} - 14}{\mathrm{x} + 2}\)
• Strategy: Factor the numerator to look for common factors with the denominator
• Goal: Cancel any common factors to simplify

2. SIMPLIFY by factoring the numerator

• Factor \(\mathrm{x}^2 - 5\mathrm{x} - 14\) using the standard method:
  • Need two numbers that multiply to -14 and add to -5
  • Test factor pairs of -14: \((1,-14), (-1,14), (2,-7), (-2,7)\)
  • Check: \(2 + (-7) = -5\) ✓ and \((2)(-7) = -14\) ✓
• Therefore: \(\mathrm{x}^2 - 5\mathrm{x} - 14 = (\mathrm{x} + 2)(\mathrm{x} - 7)\)

3. SIMPLIFY by canceling common factors

• Substitute the factored form:
  \(\frac{\mathrm{x}^2 - 5\mathrm{x} - 14}{\mathrm{x} + 2} = \frac{(\mathrm{x} + 2)(\mathrm{x} - 7)}{\mathrm{x} + 2}\)
• Cancel the common factor \((\mathrm{x} + 2)\):
  \(= \mathrm{x} - 7\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Incorrectly factoring the quadratic expression

Students often struggle to find the correct factor pair for \(\mathrm{x}^2 - 5\mathrm{x} - 14\). Common mistakes include:

• Using factors that multiply correctly but add incorrectly (like -2 and 7, which multiply to -14 but add to 5, not -5)
• Getting confused with signs and writing factors like \((\mathrm{x} - 2)(\mathrm{x} + 7)\) instead of \((\mathrm{x} + 2)(\mathrm{x} - 7)\)
• Forgetting to check their factoring by expanding back

This leads to an incorrect simplified expression and may cause them to select Choice B \((\mathrm{x} - 9)\), Choice C \((\mathrm{x} + 7)\), or Choice D \((\mathrm{x} + 9)\).

Second Most Common Error:

Poor INFER reasoning: Not recognizing that factoring is the key strategy

Some students attempt to use polynomial long division or other complex methods instead of recognizing that factoring the numerator is the most efficient approach. This can lead to calculation errors or getting stuck partway through.

This causes them to abandon systematic solution and guess among the answer choices.

The Bottom Line:

Success on this problem depends entirely on your factoring skills. If you can correctly factor quadratic expressions, rational expression simplification becomes straightforward. The key insight is recognizing when the denominator appears as a factor in the numerator.