Which of the following expressions is equivalent to the result of subtracting \(\mathrm{(x^3 + 4x^2 - 6x)}\) from \(\mathrm{(3x^3 -...
GMAT Advanced Math : (Adv_Math) Questions
Which of the following expressions is equivalent to the result of subtracting \(\mathrm{(x^3 + 4x^2 - 6x)}\) from \(\mathrm{(3x^3 - 2x^2 + 5)}\)?
1. TRANSLATE the problem information
- Given: Subtract \(\mathrm{(x^3 + 4x^2 - 6x)}\) from \(\mathrm{(3x^3 - 2x^2 + 5)}\)
- What this means: \(\mathrm{(3x^3 - 2x^2 + 5) - (x^3 + 4x^2 - 6x)}\)
- Key insight: "Subtract A from B" means B - A, not A - B
2. SIMPLIFY by distributing the negative sign
- \(\mathrm{(3x^3 - 2x^2 + 5) - (x^3 + 4x^2 - 6x)}\)
- Distribute the negative to each term in the second polynomial:
- \(\mathrm{= 3x^3 - 2x^2 + 5 - x^3 - 4x^2 + 6x}\)
- Notice: \(\mathrm{-(-6x)}\) becomes \(\mathrm{+6x}\)
3. SIMPLIFY by combining like terms
- Group terms by degree:
- \(\mathrm{x^3}\) terms: \(\mathrm{3x^3 - x^3 = 2x^3}\)
- \(\mathrm{x^2}\) terms: \(\mathrm{-2x^2 - 4x^2 = -6x^2}\)
- \(\mathrm{x}\) terms: \(\mathrm{0 + 6x = 6x}\) (first polynomial had no x term)
- Constants: \(\mathrm{+5}\)
- Final result: \(\mathrm{2x^3 - 6x^2 + 6x + 5}\)
Answer: B. \(\mathrm{2x^3 - 6x^2 + 6x + 5}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Setting up the subtraction backwards as \(\mathrm{(x^3 + 4x^2 - 6x) - (3x^3 - 2x^2 + 5)}\) instead of the correct order.
The phrase "subtract A from B" confuses many students who interpret it as "A - B" rather than "B - A". This backwards setup leads to \(\mathrm{4x^3 + 2x^2 - 6x - 5}\), which doesn't match any answer choice, causing them to get confused and guess.
Second Most Common Error:
Poor SIMPLIFY execution: Making sign errors when distributing the negative sign, particularly forgetting that \(\mathrm{-(-6x) = +6x}\).
Students often write: \(\mathrm{3x^3 - 2x^2 + 5 - x^3 - 4x^2 - 6x}\), keeping the wrong sign on the x term. This leads to \(\mathrm{2x^3 - 6x^2 - 6x + 5}\), which matches Choice A.
The Bottom Line:
This problem tests whether students can correctly interpret subtraction language and carefully track signs through distribution. The key challenge is remembering that subtracting a negative term creates a positive result.