Which expression is equivalent to (x^2 + 3x - 40)/(x - 5) for all values of x for which the...

1. INFER the solution strategy

• Looking at \(\frac{\mathrm{x^2 + 3x - 40}}{\mathrm{x - 5}}\), I notice the denominator \(\mathrm{(x - 5)}\) suggests that \(\mathrm{(x - 5)}\) might be a factor of the numerator
• Strategy: Factor the numerator and look for common factors to cancel

2. SIMPLIFY by factoring the numerator

• For \(\mathrm{x^2 + 3x - 40}\), I need two numbers that:
  • Multiply to give \(\mathrm{-40}\)
  • Add to give \(\mathrm{3}\)
• Testing factor pairs: \(\mathrm{8 \times (-5) = -40}\) and \(\mathrm{8 + (-5) = 3}\) ✓
• Therefore: \(\mathrm{x^2 + 3x - 40 = (x + 8)(x - 5)}\)

3. SIMPLIFY by substituting and canceling

• Original expression: \(\frac{\mathrm{x^2 + 3x - 40}}{\mathrm{x - 5}}\)
• After factoring: \(\frac{\mathrm{(x + 8)(x - 5)}}{\mathrm{(x - 5)}}\)
• Cancel the common factor \(\mathrm{(x - 5)}\): \(\mathrm{x + 8}\)

Answer: D) \(\mathrm{x + 8}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students struggle to find the correct factor pair for \(\mathrm{x^2 + 3x - 40}\). They might try random combinations without systematic thinking, leading to incorrect factorizations like \(\mathrm{(x + 10)(x - 4)}\) or \(\mathrm{(x + 20)(x - 2)}\). With wrong factors, they can't properly cancel terms and may resort to guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize that the denominator \(\mathrm{(x - 5)}\) provides a clue about what factors to look for in the numerator. Instead, they might attempt polynomial long division or other complicated approaches, making the problem much harder than necessary and potentially running out of time.

The Bottom Line:

Success on this problem requires recognizing the connection between the denominator and potential factors of the numerator, then systematically finding the correct factor pair through organized trial.