Which expression is a factor of 6y^2 + 24y + 18?

1. INFER the approach needed

• Key insight: To find a factor, I need to factor the polynomial completely first
• Strategy: Look for the greatest common factor (GCF) of all terms, then factor further if possible

2. SIMPLIFY by finding the GCF

• Look at coefficients: 6, 24, 18
  • \(\mathrm{6 = 2 \times 3}\)
  • \(\mathrm{24 = 2^3 \times 3}\)
  • \(\mathrm{18 = 2 \times 3^2}\)
  • GCF = 6
• Factor out 6: \(\mathrm{6y^2 + 24y + 18 = 6(y^2 + 4y + 3)}\)

3. SIMPLIFY the remaining quadratic

• Factor \(\mathrm{y^2 + 4y + 3}\)
• Need two numbers that multiply to 3 and add to 4
• Those numbers are 3 and 1
• So \(\mathrm{y^2 + 4y + 3 = (y + 3)(y + 1)}\)

4. INFER the complete factorization

• Complete factorization: \(\mathrm{6y^2 + 24y + 18 = 6(y + 3)(y + 1)}\)
• Since \(\mathrm{6 = 2 \times 3}\), the factors are: \(\mathrm{6, 2, 3, (y + 3), (y + 1)}\), and combinations

5. INFER which answer choice is correct

• Looking at the choices: 3 is a factor of 6, which makes it a factor of the entire expression
• The other choices \(\mathrm{(4y, 24y, 6y^2)}\) are terms within the original expression, not factors

Answer: A (3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse terms within the polynomial with factors of the polynomial.

They see \(\mathrm{4y, 24y,}\) or \(\mathrm{6y^2}\) as parts of the expression and think these must be factors. However, a factor must divide the entire expression evenly, while these are just individual terms. This conceptual confusion about what constitutes a factor versus what constitutes a term leads them to select Choice B (4y), Choice C (24y), or Choice D \(\mathrm{(6y^2)}\).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students attempt to factor but don't complete the process systematically.

They might recognize they need to factor but either don't find the GCF correctly or stop partway through the factoring process. Without the complete factorization, they can't identify all the true factors and end up guessing among the answer choices.

The Bottom Line:

This problem tests whether students understand the fundamental difference between the terms that make up a polynomial expression and the factors that divide the entire expression. Success requires both systematic algebraic manipulation and clear conceptual understanding of what "factor" means.