What is one of the solutions to the given equation?\(\mathrm{3x(x - 4)(x + 5) = 0}\)

1. INFER the solving strategy

• Given: \(\mathrm{3x(x - 4)(x + 5) = 0}\)
• Key insight: This equation is already factored, so we can apply the zero product property
• Strategy: Set each factor equal to zero and solve

2. APPLY the zero product property

• If a product equals zero, then at least one factor must be zero
• This gives us three separate equations:
  • \(\mathrm{3x = 0}\)
  • \(\mathrm{x - 4 = 0}\)
  • \(\mathrm{x + 5 = 0}\)

3. SIMPLIFY by solving each equation

• From \(\mathrm{3x = 0}\): Divide both sides by 3 → \(\mathrm{x = 0}\)
• From \(\mathrm{x - 4 = 0}\): Add 4 to both sides → \(\mathrm{x = 4}\)
• From \(\mathrm{x + 5 = 0}\): Subtract 5 from both sides → \(\mathrm{x = -5}\)

4. TRANSLATE solutions to answer choice

• Our solutions are: \(\mathrm{x = 0}\), \(\mathrm{x = 4}\), and \(\mathrm{x = -5}\)
• Looking at the choices: A. -4, B. 0, C. 3, D. 5
• Only \(\mathrm{x = 0}\) appears in the answer choices

Answer: B. 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to use the zero product property. Instead, they think the solutions are the numbers that appear in the factors themselves: -4 (from \(\mathrm{x - 4}\)), 3 (the coefficient), and 5 (from \(\mathrm{x + 5}\)). This leads them to look for these values in the answer choices and may select Choice A (-4) or Choice D (5) without actually solving.

Second Most Common Error:

Poor SIMPLIFY execution: Students apply the zero product property correctly but make sign errors when solving the linear equations. For example, from \(\mathrm{x + 5 = 0}\), they might incorrectly get \(\mathrm{x = 5}\) instead of \(\mathrm{x = -5}\). This could lead them to select Choice D (5) thinking it's a valid solution.

The Bottom Line:

This problem tests whether students can recognize when to apply the zero product property and then execute the algebraic steps correctly. The key insight is that factored form immediately signals "use zero product property" - don't try to expand or do other manipulations first.