Question:A polynomial P is defined by P = 12y^2 + 30.Which expression is equivalent to P?
GMAT Advanced Math : (Adv_Math) Questions
A polynomial \(\mathrm{P}\) is defined by \(\mathrm{P = 12y^2 + 30}\).
Which expression is equivalent to \(\mathrm{P}\)?
1. INFER the problem type and strategy
- This is asking for an equivalent expression to \(\mathrm{P = 12y^2 + 30}\)
- Since we want an equivalent form and the answer choices show factored expressions, this is a factoring problem
- Strategy: Find the greatest common factor (GCF) of all terms
2. SIMPLIFY to find the GCF
- For the numerical coefficients:
- \(\mathrm{12 = 2^2 \times 3}\)
- \(\mathrm{30 = 2 \times 3 \times 5}\)
- GCF of coefficients = \(\mathrm{2 \times 3 = 6}\)
- For the variable parts:
- First term: \(\mathrm{y^2}\)
- Second term: no y
- GCF of variables = 1 (no common variable factor)
- Overall GCF = 6
3. SIMPLIFY by factoring out the GCF
- Factor out 6 from each term:
\(\mathrm{12y^2 + 30 = 6(2y^2) + 6(5) = 6(2y^2 + 5)}\)
4. Verify the answer
- Distribute back: \(\mathrm{6(2y^2 + 5) = 12y^2 + 30}\) ✓
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Incorrectly including y as part of the GCF when the constant term has no y factor.
Students see \(\mathrm{y^2}\) in the first term and mistakenly think y should be factored out from both terms. They attempt: \(\mathrm{12y^2 + 30 = 6y(2y + ?)}\) but then get stuck because 30 doesn't contain y. Some force it anyway and write \(\mathrm{6y(2y + 5)}\), not realizing this gives \(\mathrm{12y^2 + 30y}\) instead of \(\mathrm{12y^2 + 30}\).
This may lead them to select Choice C (\(\mathrm{6y(2y + 5)}\)).
Second Most Common Error:
Poor SIMPLIFY execution: Making arithmetic errors when dividing the constant term by the GCF.
Students correctly identify 6 as the GCF but make a calculation error: \(\mathrm{30 \div 6 = 15}\) instead of \(\mathrm{30 \div 6 = 5}\). This gives them \(\mathrm{6(2y^2 + 15)}\).
This may lead them to select Choice B (\(\mathrm{6(2y^2 + 15)}\)).
The Bottom Line:
Success requires systematically finding the GCF by looking at what ALL terms have in common, not just focusing on one term. The constant term 30 has no variable factors, so y cannot be part of the GCF.