Which expression is equivalent to \((7\mathrm{x}^3 + 4\mathrm{x}^2 - 3) - (2\mathrm{x}^3 - 6\mathrm{x} + 8)\)?

1. TRANSLATE the subtraction into distribution

• Given: \((7\mathrm{x}^3 + 4\mathrm{x}^2 - 3) - (2\mathrm{x}^3 - 6\mathrm{x} + 8)\)
• Key insight: Subtracting a polynomial means adding its opposite, so distribute the negative sign to every term in the second polynomial

2. SIMPLIFY by distributing the negative sign

• \((7\mathrm{x}^3 + 4\mathrm{x}^2 - 3) - (2\mathrm{x}^3 - 6\mathrm{x} + 8)\)
• \(= 7\mathrm{x}^3 + 4\mathrm{x}^2 - 3 - 2\mathrm{x}^3 + 6\mathrm{x} - 8\)
• Notice: \(-(2\mathrm{x}^3) = -2\mathrm{x}^3\), \(-(-6\mathrm{x}) = +6\mathrm{x}\), \(-(8) = -8\)

3. SIMPLIFY by grouping and combining like terms

• Group like terms together:
  • \(\mathrm{x}^3\) terms: \(7\mathrm{x}^3 - 2\mathrm{x}^3 = 5\mathrm{x}^3\)
  • \(\mathrm{x}^2\) terms: \(4\mathrm{x}^2\) (only one term)
  • \(\mathrm{x}\) terms: \(6\mathrm{x}\) (only one term)
  • constants: \(-3 - 8 = -11\)
• Final expression: \(5\mathrm{x}^3 + 4\mathrm{x}^2 + 6\mathrm{x} - 11\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when distributing the negative sign, especially with the \(-6\mathrm{x}\) term.

Students often write \(-(2\mathrm{x}^3 - 6\mathrm{x} + 8)\) as \(-2\mathrm{x}^3 - 6\mathrm{x} - 8\) instead of \(-2\mathrm{x}^3 + 6\mathrm{x} - 8\), forgetting that subtracting a negative gives a positive. This leads to getting \(-6\mathrm{x}\) instead of \(+6\mathrm{x}\) in their final answer.

This may lead them to select Choice A \((5\mathrm{x}^3 + 4\mathrm{x}^2 - 6\mathrm{x} - 5)\).

Second Most Common Error:

Poor SIMPLIFY reasoning: Incorrectly combining the constant terms.

Students correctly handle the variable terms but make arithmetic errors with the constants, calculating \(-3 - 8\) as \(-5\) instead of \(-11\), or sometimes even as \(+5\).

This may lead them to select Choice A or Choice C \((5\mathrm{x}^3 + 4\mathrm{x}^2 + 6\mathrm{x} + 5)\).

The Bottom Line:

This problem tests careful attention to signs during distribution and systematic organization when combining like terms. The key is treating subtraction as "adding the opposite" and being methodical about sign changes.