Which of the following is equivalent to the expression 18 - 3y^2?

1. INFER the solution strategy

• This is an algebraic factoring problem
• I need to find a common factor that can be pulled out from both terms
• The key insight: factoring is the reverse of distribution

2. INFER which common factor to use

• Look at the coefficients: \(18\) and \(3\)
• Find their greatest common factor: \(\mathrm{GCF}(18, 3) = 3\)
• Since both terms have this factor, I can factor out \(3\)

3. SIMPLIFY by factoring out the GCF

• Factor out \(3\) from each term:
  • \(18 ÷ 3 = 6\)
  • \(3\mathrm{y}^2 ÷ 3 = \mathrm{y}^2\)
• Write the factored form: \(18 - 3\mathrm{y}^2 = 3(6 - \mathrm{y}^2)\)

4. SIMPLIFY through verification (optional but recommended)

• Expand choice (A): \(3(6 - \mathrm{y}^2) = 18 - 3\mathrm{y}^2\) ✓
• This matches our original expression

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that factoring requires finding the GCD first, or they guess which factor to use instead of systematically finding it.

They might factor out \(6\) instead of \(3\) (since \(6\) divides \(18\)), leading to something like \(6(3 - ?\mathrm{y}^2)\), but then get confused because \(6\) doesn't divide \(3\mathrm{y}^2\) evenly. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify \(3\) as the GCF but make sign errors when factoring.

They might write \(3(6 + \mathrm{y}^2)\) instead of \(3(6 - \mathrm{y}^2)\), forgetting that \(-3\mathrm{y}^2 ÷ 3 = -\mathrm{y}^2\), not \(+\mathrm{y}^2\). This may lead them to select Choice B (\(3(6 + \mathrm{y}^2)\)).

The Bottom Line:

This problem tests whether students can systematically factor expressions rather than just guess-and-check. The key insight is recognizing that factoring means finding what's truly common to both terms, not just what divides one of them.