The area of a rectangular garden is given by the expression 12x + 18. Which of the following represents an...

1. INFER what the problem is asking

• Given: Area expression \(\mathrm{12x + 18}\)
• Need: Equivalent expression 'written as a product'
• This tells us we need to factor the expression (rewrite as multiplication)

2. SIMPLIFY by finding the greatest common factor

• Look at each term: \(\mathrm{12x}\) and \(\mathrm{18}\)
• Find factors of the coefficients:
  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Factors of 18: 1, 2, 3, 6, 9, 18
• The GCF is 6 (largest number that divides both)

3. SIMPLIFY by factoring out the GCF

• Factor out 6 from each term:
  • \(\mathrm{12x \div 6 = 2x}\)
  • \(\mathrm{18 \div 6 = 3}\)
• Write as: \(\mathrm{12x + 18 = 6(2x + 3)}\)

4. Verify your answer

• Check: \(\mathrm{6(2x + 3) = 6(2x) + 6(3) = 12x + 18}\) ✓

Answer: A) \(\mathrm{6(2x + 3)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students find the correct GCF of 6 but make errors when dividing the terms by the factor. They might forget to divide the constant term (18) or the coefficient (12) by 6, leading to expressions like \(\mathrm{6(2x + 18)}\) or \(\mathrm{6(12x + 3)}\).

This may lead them to select Choice B (\(\mathrm{6(2x + 18)}\)) or Choice C (\(\mathrm{6(12x + 3)}\)).

Second Most Common Error:

Poor INFER reasoning about factor selection: Students might choose a different common factor (like 2 or 3) instead of the greatest common factor, or use an incorrect factor altogether like 12. When using 12 as the factor, they incorrectly determine that \(\mathrm{18 \div 12 = 3}\), leading to \(\mathrm{12(x + 3)}\).

This may lead them to select Choice D (\(\mathrm{12(x + 3)}\)).

The Bottom Line:

This problem tests whether students can systematically apply the factoring process. Success requires both finding the correct GCF and then accurately dividing both terms by that factor—missing either step leads directly to one of the distractor answers.