Which of the following expressions is equivalent to 12x^3 - 6x^2?

1. INFER the problem strategy

• This expression has two terms with common factors
• Strategy: Find the greatest common factor (GCF) and factor it out
• This will give us an equivalent expression in factored form

2. SIMPLIFY by finding the GCF of coefficients

• Coefficients are 12 and 6
• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 6: 1, 2, 3, 6
• GCF of coefficients = 6

3. SIMPLIFY by finding the GCF of variables

• Variable parts are \(\mathrm{x^3}\) and \(\mathrm{x^2}\)
• The GCF is the lowest power that appears: \(\mathrm{x^2}\)
• Overall GCF = \(\mathrm{6x^2}\)

4. SIMPLIFY by factoring out the GCF

• Divide each term by \(\mathrm{6x^2}\):
  • First term: \(\mathrm{12x^3 ÷ 6x^2 = 2x}\)
  • Second term: \(\mathrm{-6x^2 ÷ 6x^2 = -1}\)
• Factored form: \(\mathrm{6x^2(2x - 1)}\)

5. Verify the answer

• Check: \(\mathrm{6x^2(2x - 1) = 6x^2 · 2x + 6x^2 · (-1) = 12x^3 - 6x^2}\) ✓

Answer: C) \(\mathrm{6x^2(2x - 1)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly identify \(\mathrm{6x^2}\) as the GCF but make a sign error when factoring out the second term.

When dividing \(\mathrm{-6x^2}\) by \(\mathrm{6x^2}\), they might forget the negative sign or mishandle it, getting +1 instead of -1. This leads them to write \(\mathrm{6x^2(2x + 1)}\) instead of \(\mathrm{6x^2(2x - 1)}\).

This may lead them to select Choice D (\(\mathrm{6x^2(2x + 1)}\))

Second Most Common Error:

Poor INFER reasoning: Students don't recognize this as a factoring problem and instead try to combine like terms or use other inappropriate strategies.

Without a clear factoring strategy, they might look for terms that "match" the original expression and randomly select among the choices.

This leads to confusion and guessing.

The Bottom Line:

Success depends on systematic GCF identification and careful attention to signs during the factoring process. The key insight is recognizing that factoring creates an equivalent expression, not a simplified one.