Which of the following is equivalent to \((3\mathrm{x}^3 + 2\mathrm{x})(\mathrm{x}^3 - \mathrm{x}) + (\mathrm{x}^3 + \mathrm{x})(\mathrm{x}^3 - \mathr...
GMAT Advanced Math : (Adv_Math) Questions
Which of the following is equivalent to \((3\mathrm{x}^3 + 2\mathrm{x})(\mathrm{x}^3 - \mathrm{x}) + (\mathrm{x}^3 + \mathrm{x})(\mathrm{x}^3 - \mathrm{x})\)?
\(4\mathrm{x}^6 + \mathrm{x}^4 - 3\mathrm{x}^2\)
\(4\mathrm{x}^6 - \mathrm{x}^4 - 3\mathrm{x}^2\)
\(4\mathrm{x}^6 + \mathrm{x}^4 - 2\mathrm{x}^2\)
\(4\mathrm{x}^6 - \mathrm{x}^4 + 3\mathrm{x}^2\)
\(3\mathrm{x}^6 - \mathrm{x}^4 - 3\mathrm{x}^2\)
1. INFER the most efficient approach
Looking at: \((3x^3 + 2x)(x^3 - x) + (x^3 + x)(x^3 - x)\)
- Key insight: Both terms contain the identical factor \((x^3 - x)\)
- Strategy: Factor out this common binomial rather than expanding everything first
- This will make our work much simpler!
2. SIMPLIFY by factoring out the common factor
- Factor out \((x^3 - x)\): \([(3x^3 + 2x) + (x^3 + x)](x^3 - x)\)
- Combine the terms in brackets: \((3x^3 + 2x) + (x^3 + x) = 4x^3 + 3x\)
- Result: \((4x^3 + 3x)(x^3 - x)\)
3. SIMPLIFY by expanding the factored expression
- Use distributive property: \((4x^3 + 3x)(x^3 - x)\)
- First term: \(4x^3 \cdot x^3 = 4x^6\) and \(4x^3 \cdot (-x) = -4x^4\)
- Second term: \(3x \cdot x^3 = 3x^4\) and \(3x \cdot (-x) = -3x^2\)
- Combine: \(4x^6 - 4x^4 + 3x^4 - 3x^2\)
- Final simplification: \(4x^6 - x^4 - 3x^2\)
Answer: B (\(4x^6 - x^4 - 3x^2\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Not recognizing the common factor and trying to expand each product separately first.
Students expand \((3x^3 + 2x)(x^3 - x) = 3x^6 - 3x^4 + 2x^4 - 2x^2\) and \((x^3 + x)(x^3 - x) = x^6 - x^4 + x^4 - x^2\) separately, then try to combine. While this eventually leads to the correct answer, it's much more work and creates more opportunities for calculation errors.
This approach often leads to arithmetic mistakes that result in selecting incorrect answer choices like Choice A (\(4x^6 + x^4 - 3x^2\)) or Choice C (\(4x^6 + x^4 - 2x^2\)).
Second Most Common Error:
Poor SIMPLIFY execution: Making sign errors during the expansion of \((4x^3 + 3x)(x^3 - x)\).
Students correctly factor but then mess up the signs when expanding: forgetting that \(4x^3 \times (-x) = -4x^4\) or that \(3x \times (-x) = -3x^2\). They might also incorrectly combine \(-4x^4 + 3x^4\), getting \(+x^4\) instead of \(-x^4\).
This may lead them to select Choice A (\(4x^6 + x^4 - 3x^2\)) when they get the wrong sign on the \(x^4\) term.
The Bottom Line:
This problem rewards students who can spot patterns and factor strategically rather than just applying brute-force expansion. The key insight is recognizing the repeated factor \((x^3 - x)\) immediately.
\(4\mathrm{x}^6 + \mathrm{x}^4 - 3\mathrm{x}^2\)
\(4\mathrm{x}^6 - \mathrm{x}^4 - 3\mathrm{x}^2\)
\(4\mathrm{x}^6 + \mathrm{x}^4 - 2\mathrm{x}^2\)
\(4\mathrm{x}^6 - \mathrm{x}^4 + 3\mathrm{x}^2\)
\(3\mathrm{x}^6 - \mathrm{x}^4 - 3\mathrm{x}^2\)