Which of the following expressions is equivalent to \((3\mathrm{x}^2\mathrm{y} - 5\mathrm{xy} + 2) - (\mathrm{x}^2\mathrm{y} + 2\mathrm{xy} - 6)\)?

1. INFER the solution strategy

• When subtracting polynomials, we distribute the negative sign to every term in the second polynomial
• Then we combine like terms by adding/subtracting their coefficients

2. SIMPLIFY by distributing the negative sign

• Original: \((3\mathrm{x}^2\mathrm{y} - 5\mathrm{xy} + 2) - (\mathrm{x}^2\mathrm{y} + 2\mathrm{xy} - 6)\)
• Distribute the negative: \(3\mathrm{x}^2\mathrm{y} - 5\mathrm{xy} + 2 - \mathrm{x}^2\mathrm{y} - 2\mathrm{xy} + 6\)
• Notice that \(-(−6)\) becomes \(+6\)

3. SIMPLIFY by grouping like terms

• Group \(\mathrm{x}^2\mathrm{y}\) terms: \((3\mathrm{x}^2\mathrm{y} - \mathrm{x}^2\mathrm{y})\)
• Group \(\mathrm{xy}\) terms: \((-5\mathrm{xy} - 2\mathrm{xy})\)
• Group constants: \((2 + 6)\)

4. SIMPLIFY by combining coefficients

• \(\mathrm{x}^2\mathrm{y}\) terms: \(3 - 1 = 2\) → \(2\mathrm{x}^2\mathrm{y}\)
• \(\mathrm{xy}\) terms: \(-5 - 2 = -7\) → \(-7\mathrm{xy}\)
• Constants: \(2 + 6 = 8\)

Answer: \(2\mathrm{x}^2\mathrm{y} - 7\mathrm{xy} + 8\) (Choice B)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Not properly distributing the negative sign to the constant term

Students correctly distribute the negative to \(\mathrm{x}^2\mathrm{y}\) and \(2\mathrm{xy}\), but forget that \(-(−6) = +6\). They keep it as \(-6\), leading to:

\(2\mathrm{x}^2\mathrm{y} - 7\mathrm{xy} + 2 - 6 = 2\mathrm{x}^2\mathrm{y} - 7\mathrm{xy} - 4\)

This leads them to select Choice A (\(2\mathrm{x}^2\mathrm{y} - 7\mathrm{xy} - 4\))

Second Most Common Error:

Weak SIMPLIFY execution: Arithmetic errors when combining the \(\mathrm{xy}\) terms

Students might incorrectly compute \(-5\mathrm{xy} - 2\mathrm{xy}\) as \(-3\mathrm{xy}\) instead of \(-7\mathrm{xy}\), possibly thinking \(-5 + 2 = -3\). This gives:

\(2\mathrm{x}^2\mathrm{y} - 3\mathrm{xy} + 8\)

This leads them to select Choice C (\(2\mathrm{x}^2\mathrm{y} - 3\mathrm{xy} + 8\))

The Bottom Line:

The key challenge is careful attention to sign changes when distributing the negative, especially with the constant term where a negative times negative becomes positive.