Which of the following expressions is equivalent to the difference when 2m^2 + 11m + 8 is subtracted from m^3...
GMAT Advanced Math : (Adv_Math) Questions
Which of the following expressions is equivalent to the difference when \(2\mathrm{m}^2 + 11\mathrm{m} + 8\) is subtracted from \(\mathrm{m}^3 + 6\mathrm{m}^2 + 5\mathrm{m} + 20\)?
1. TRANSLATE the problem language
- Given: Find the difference when \((2\mathrm{m}^2 + 11\mathrm{m} + 8)\) is subtracted from \((\mathrm{m}^3 + 6\mathrm{m}^2 + 5\mathrm{m} + 20)\)
- This translates to: \((\mathrm{m}^3 + 6\mathrm{m}^2 + 5\mathrm{m} + 20) - (2\mathrm{m}^2 + 11\mathrm{m} + 8)\)
- Key insight: "A is subtracted from B" means B - A
2. SIMPLIFY by distributing the negative sign
- \((\mathrm{m}^3 + 6\mathrm{m}^2 + 5\mathrm{m} + 20) - (2\mathrm{m}^2 + 11\mathrm{m} + 8)\)
- = \(\mathrm{m}^3 + 6\mathrm{m}^2 + 5\mathrm{m} + 20 - 2\mathrm{m}^2 - 11\mathrm{m} - 8\)
- Remember: Subtracting each term changes all signs in the second polynomial
3. SIMPLIFY by combining like terms
- Group terms by degree:
- \(\mathrm{m}^3\) terms: \(\mathrm{m}^3\)
- \(\mathrm{m}^2\) terms: \(6\mathrm{m}^2 - 2\mathrm{m}^2 = 4\mathrm{m}^2\)
- \(\mathrm{m}\) terms: \(5\mathrm{m} - 11\mathrm{m} = -6\mathrm{m}\)
- constant terms: \(20 - 8 = 12\)
- Final result: \(\mathrm{m}^3 + 4\mathrm{m}^2 - 6\mathrm{m} + 12\)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret "when A is subtracted from B" and set up the problem as A - B instead of B - A.
This backward setup would give them \((2\mathrm{m}^2 + 11\mathrm{m} + 8) - (\mathrm{m}^3 + 6\mathrm{m}^2 + 5\mathrm{m} + 20)\), leading to a completely different polynomial with negative leading coefficient. While this exact result isn't among the choices, the confusion often leads to randomly selecting an answer.
Second Most Common Error:
Poor SIMPLIFY execution: Students make sign errors when distributing the negative sign or combining like terms.
Common mistakes include:
- Getting \(6\mathrm{m}^2 + 2\mathrm{m}^2 = 8\mathrm{m}^2\) instead of \(6\mathrm{m}^2 - 2\mathrm{m}^2 = 4\mathrm{m}^2\) (leads to Choice D)
- Getting \(5\mathrm{m} + 11\mathrm{m} = 16\mathrm{m}\) or miscalculating to get \(+6\mathrm{m}\) instead of \(-6\mathrm{m}\) (leads to Choice B)
- Getting \(20 + 8 = 28\) instead of \(20 - 8 = 12\) (leads to Choice C)
The Bottom Line:
The confusing "subtracted from" language combined with multiple opportunities for sign errors makes this problem challenging despite using only basic polynomial operations.