Which expression is equivalent to 12x + 27?

1. INFER the approach

• We need to find which expression equals \(\mathrm{12x + 27}\)
• Two strategies:
  • Factor \(\mathrm{12x + 27}\) and match it to an answer choice
  • Expand each answer choice and see which equals \(\mathrm{12x + 27}\)

2. SIMPLIFY by factoring the original expression

• Look for the greatest common factor of 12x and 27:
  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Factors of 27: 1, 3, 9, 27
  • Greatest common factor: 3

• Factor out the 3:
  • \(\mathrm{12x + 27 = 3(4x) + 3(9) = 3(4x + 9)}\)

3. Match to answer choices

• Our factored form is \(\mathrm{3(4x + 9)}\)
• This exactly matches Choice C

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when factoring or when checking answer choices by expansion.

For example, they might incorrectly think \(\mathrm{12x + 27 = 3(4x + 24)}\) because they divide 27 by 3 incorrectly, or they might expand \(\mathrm{3(4x + 9)}\) as \(\mathrm{12x + 12}\) by forgetting to multiply \(\mathrm{3 \times 9}\). This leads them to select Choice D (\(\mathrm{3(9x + 24)}\)) or causes confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize they need to factor the expression or systematically check the answer choices.

Instead, they might try to guess based on which numbers "look right" or get overwhelmed by not having a clear strategy. This leads to abandoning systematic solution and guessing randomly.

The Bottom Line:

This problem tests whether students can systematically apply the distributive property in both directions - factoring an expression and expanding to check their work. Success requires careful arithmetic combined with strategic thinking.