Which of the following expressions is equivalent to the expression 18x^3 + 12x^2 - 3x?\(3(6\mathrm{x}^3 + 4\mathrm{x}^2 - 2\mathrm{x})\)\(3\mathrm{x}(...
GMAT Advanced Math : (Adv_Math) Questions
Which of the following expressions is equivalent to the expression \(18\mathrm{x}^3 + 12\mathrm{x}^2 - 3\mathrm{x}\)?
- \(3(6\mathrm{x}^3 + 4\mathrm{x}^2 - 2\mathrm{x})\)
- \(3\mathrm{x}(6\mathrm{x}^2 + 4\mathrm{x} - 1)\)
- \(6\mathrm{x}(3\mathrm{x}^2 + 2\mathrm{x})\)
- \(\mathrm{x}(18\mathrm{x}^2 + 12\mathrm{x} + 3)\)
1. INFER the problem strategy
- Given: \(\mathrm{18x^3 + 12x^2 - 3x}\)
- This is asking for an equivalent expression, which means we need to factor
- Strategy: Find the greatest common factor (GCF) and factor it out
2. SIMPLIFY to find the GCF of coefficients
- Coefficients: \(\mathrm{18, 12, -3}\)
- Find prime factorizations:
- \(\mathrm{18 = 2 \times 3^2}\)
- \(\mathrm{12 = 2^2 \times 3}\)
- \(\mathrm{3 = 3}\)
- GCF of coefficients = 3
3. SIMPLIFY to find the GCF of variables
- Variable terms: \(\mathrm{x^3, x^2, x}\)
- GCF = \(\mathrm{x^1 = x}\) (take the lowest power)
4. SIMPLIFY by factoring out the overall GCF
- Overall GCF = \(\mathrm{3x}\)
- Divide each term by \(\mathrm{3x}\):
- \(\mathrm{18x^3 \div (3x) = 6x^2}\)
- \(\mathrm{12x^2 \div (3x) = 4x}\)
- \(\mathrm{-3x \div (3x) = -1}\)
- Factored form: \(\mathrm{3x(6x^2 + 4x - 1)}\)
5. INFER the correct answer choice
- Compare \(\mathrm{3x(6x^2 + 4x - 1)}\) with the given options
- This matches choice (B)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY skill: Students factor out an incomplete GCF, such as factoring out only 3 instead of \(\mathrm{3x}\).
When students only factor out 3, they get \(\mathrm{3(6x^3 + 4x^2 - x)}\), which matches the format of choice (A) except for the coefficient of the last term. This partial factoring error leads to confusion when trying to match answer choices, causing them to either select Choice A incorrectly or get stuck and guess.
Second Most Common Error:
Poor SIMPLIFY execution: Students make computational errors when dividing terms by the GCF.
For example, they might correctly identify \(\mathrm{3x}\) as the GCF but then make errors like \(\mathrm{-3x \div 3x = -3}\) instead of -1, or \(\mathrm{12x^2 \div 3x = 4x^2}\) instead of \(\mathrm{4x}\). These division errors create expressions that don't match any of the answer choices, leading to confusion and random guessing.
The Bottom Line:
This problem requires systematic execution of the factoring process. Students must correctly identify the complete GCF (including both numerical and variable parts) and then accurately perform the division for each term. Rushing through either step typically derails the entire solution.