For which value of x is the expression \(\frac{(\mathrm{x} + 3)(\mathrm{x} - 5)}{(\mathrm{x} - 1)(\mathrm{x} + 6)}\) equal to zero?

1. INFER the key principle

• A rational expression equals zero when:
  • Its numerator equals zero, AND
  • Its denominator does not equal zero

• This means we need to find where \((\mathrm{x} + 3)(\mathrm{x} - 5) = 0\), but \((\mathrm{x} - 1)(\mathrm{x} + 6) \neq 0\)

2. SIMPLIFY to find potential solutions

• Set the numerator equal to zero: \((\mathrm{x} + 3)(\mathrm{x} - 5) = 0\)
• Apply zero product property: \(\mathrm{x} + 3 = 0\) or \(\mathrm{x} - 5 = 0\)
• Solve each equation: \(\mathrm{x} = -3\) or \(\mathrm{x} = 5\)

3. APPLY CONSTRAINTS to check validity

• Check denominator: \((\mathrm{x} - 1)(\mathrm{x} + 6) \neq 0\)
• This means \(\mathrm{x} \neq 1\) and \(\mathrm{x} \neq -6\)
• For \(\mathrm{x} = -3\): \((-4)(3) = -12 \neq 0\) ✓
• For \(\mathrm{x} = 5\): \((4)(11) = 44 \neq 0\) ✓

4. APPLY CONSTRAINTS to select final answer

• Both \(\mathrm{x} = -3\) and \(\mathrm{x} = 5\) are mathematically valid
• From the given choices, only \(\mathrm{x} = -3\) appears as option (B)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the fundamental principle that fractions equal zero only when the numerator equals zero (not the denominator). They might set the entire expression equal to zero without separating numerator and denominator, or worse, set the denominator equal to zero.

This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Missing APPLY CONSTRAINTS reasoning: Students find \(\mathrm{x} = -3\) and \(\mathrm{x} = 5\) correctly but don't check which values are actually offered in the answer choices. They might get confused seeing that \(\mathrm{x} = 5\) is a valid mathematical solution but isn't listed, leading them to doubt their work.

This may lead them to select Choice (E) (6) thinking it's "close" to their \(\mathrm{x} = 5\) solution, or causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students understand the fundamental behavior of rational expressions at their zeros. The key insight is that mathematical solutions must always be filtered through the constraints of the specific question format.