Which of the following is equivalent to 2x^3 + 4?

1. INFER what the problem is asking

• The problem wants an equivalent expression to \(\mathrm{2x^3 + 4}\)
• Looking at the answer choices, they're all factored forms, so I need to factor the original expression
• This means finding the Greatest Common Factor (GCF)

2. INFER the greatest common factor

• Break down each term:
  • \(\mathrm{2x^3 = 2 \cdot x^3}\)
  • \(\mathrm{4 = 2 \cdot 2}\)
• The GCF is the largest factor that divides both terms: \(\mathrm{GCF = 2}\)

3. SIMPLIFY by factoring out the GCF

• Factor out 2 from each term:
  \(\mathrm{2x^3 + 4 = 2(x^3) + 2(2)}\)
  \(\mathrm{= 2(x^3 + 2)}\)

4. Verify the factoring

• Expand \(\mathrm{2(x^3 + 2): 2x^3 + 4}\) ✓

Answer: D. \(\mathrm{2(x^3 + 2)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly think the GCF is 4 instead of 2.

They see that 4 divides evenly into 4, but don't realize that 4 does not divide evenly into \(\mathrm{2x^3}\) (since \(\mathrm{2x^3 \div 4 = x^3/2}\), which leaves a fraction). This misconception makes them think they can factor out 4 from both terms.

This may lead them to select Choice A (\(\mathrm{4(x^3 + 4)}\)) or Choice B (\(\mathrm{4(x^3 + 2)}\)).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify that 2 is a factor of \(\mathrm{2x^3}\), but forget to factor 2 out of the constant term 4.

They write: \(\mathrm{2x^3 + 4 = 2(x^3) + 4 = 2(x^3 + 4)}\), forgetting that \(\mathrm{4 = 2\cdot 2}\).

This may lead them to select Choice C (\(\mathrm{2(x^3 + 4)}\)).

The Bottom Line:

Success on this problem requires recognizing that the GCF must divide all terms in the expression, not just some of them. The key insight is that \(\mathrm{4 = 2\cdot 2}\), so when factoring out 2, the constant becomes 2, not 4.