Which expression is equivalent to 3a^2 - 18ab^2 + 6a?

1. INFER the problem strategy

• Given: \(\mathrm{3a^2 - 18ab^2 + 6a}\)
• Strategy needed: This is asking for an equivalent expression, which suggests factoring
• Key insight: Look for a greatest common factor (GCF) that can be factored out

2. SIMPLIFY by identifying the GCF

• Examine each term's factors:
  • \(\mathrm{3a^2 = 3 \times a^2}\)
  • \(\mathrm{18ab^2 = 18 \times a \times b^2}\)
  • \(\mathrm{6a = 6 \times a}\)
• Common factors present in ALL terms: 3 and a
• Therefore, GCF = \(\mathrm{3a}\)

3. SIMPLIFY by factoring out the GCF

• Factor out \(\mathrm{3a}\) from each term:
  • \(\mathrm{3a^2 \div 3a = a}\)
  • \(\mathrm{18ab^2 \div 3a = 6b^2}\)
  • \(\mathrm{6a \div 3a = 2}\)
• Result: \(\mathrm{3a(a - 6b^2 + 2)}\)

4. Verify the answer

• SIMPLIFY by expanding: \(\mathrm{3a(a - 6b^2 + 2) = 3a^2 - 18ab^2 + 6a}\) ✓

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to find the greatest common factor of all terms. Instead, they might try to factor only some terms or look for patterns that aren't there. They might factor out just 3 (getting \(\mathrm{3(a^2 - 6ab^2 + 2a)}\)) or just a (getting \(\mathrm{a(3a - 18b^2 + 6)}\)), missing that both 3 AND a are common to all terms. This leads to confusion when none of their partial factoring matches the answer choices, causing them to guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{3a}\) should be factored out but make arithmetic errors in the division process. For example, they might incorrectly calculate \(\mathrm{18ab^2 \div 3a = 18b^2}\) instead of \(\mathrm{6b^2}\), leading them to get \(\mathrm{3a(a - 18b^2 + 2)}\). This may lead them to select Choice B (\(\mathrm{3a(a - 18b^2 + 6)}\)) after making additional computational errors.

The Bottom Line:

This problem tests whether students can systematically identify common factors across multiple terms and execute the factoring process accurately. Success requires both strategic recognition of the GCF approach and careful arithmetic throughout the factoring steps.