\((\mathrm{x} - 4)(\mathrm{x} + 2)(\mathrm{x} - 1) = 0\) What is the product of the solutions to the given equation?...

1. INFER the solving strategy

• Given: \((x - 4)(x + 2)(x - 1) = 0\)
• Key insight: When a product equals zero, at least one factor must equal zero
• Strategy: Use the zero product property to set each factor equal to zero

2. SIMPLIFY each factor equation

• Set each factor equal to zero and solve:
  • \(x - 4 = 0\) → \(x = 4\)
  • \(x + 2 = 0\) → \(x = -2\)
  • \(x - 1 = 0\) → \(x = 1\)

3. SIMPLIFY the final calculation

• The solutions are: 4, -2, and 1
• Product of solutions: \(4 \times (-2) \times 1 = -8\)

Answer: D. -8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to expand the entire expression instead of recognizing the zero product property application.

They might attempt to multiply out \((x - 4)(x + 2)(x - 1)\) first, creating a cubic equation that's much harder to solve. This leads to confusion and often causes them to abandon the systematic approach and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students find the correct solutions but make arithmetic errors when calculating the product.

They might incorrectly handle the negative sign: \(4 \times (-2) \times 1 = 8\) instead of -8, or they might find the sum instead of the product: \(4 + (-2) + 1 = 3\). This may lead them to select Choice A (8) or Choice B (3).

The Bottom Line:

This problem tests whether students can recognize when NOT to expand a factored expression and instead apply the zero product property directly. The key insight is seeing that the factored form is actually the most useful form for finding solutions.