Which of the following expressions is equivalent to \((10\mathrm{x}^3 - 4\mathrm{x}^2 + 3) - (6\mathrm{x}^3 - 7\mathrm{x}^2 + 2\mathrm{x})\)?

1. INFER the approach needed

• The problem asks us to subtract one polynomial from another
• Key insight: Subtracting a polynomial means distributing a negative sign to every term in that polynomial, then combining like terms

2. SIMPLIFY by distributing the negative sign

• Start with: \((10x^3 - 4x^2 + 3) - (6x^3 - 7x^2 + 2x)\)
• Distribute the negative to each term in the second polynomial:
  \(10x^3 - 4x^2 + 3 - 6x^3 + 7x^2 - 2x\)

3. SIMPLIFY by grouping like terms

• Rearrange to group terms with the same degree:
  \((10x^3 - 6x^3) + (-4x^2 + 7x^2) + (-2x) + 3\)

4. SIMPLIFY by combining coefficients

• x³ terms: \(10 - 6 = 4x^3\)
• x² terms: \(-4 + 7 = 3x^2\)
• x terms: \(-2x\) (stands alone)
• Constant: \(3\)

5. Write the final polynomial

• \(4x^3 + 3x^2 - 2x + 3\)

Answer: C. \(4x^3 + 3x^2 - 2x + 3\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when distributing the negative sign

Students often forget to apply the negative sign to every term in the second polynomial, especially the constant term. They might write:
\(10x^3 - 4x^2 + 3 - 6x^3 - 7x^2 + 2x\) (forgetting to change \(+2x\) to \(-2x\))

This leads to combining:
\((10x^3 - 6x^3) + (-4x^2 - 7x^2) + (2x) + 3\)
\(= 4x^3 - 11x^2 + 2x + 3\)

This may lead them to select Choice A (\(4x^3 - 11x^2 + 2x + 3\))

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors when combining coefficients

Students correctly distribute the negative sign but make calculation mistakes when combining like terms, particularly with the x² terms: \(-4 + 7\). Some might calculate this as -3 instead of +3.

This may lead them to select Choice B (\(4x^3 - 3x^2 - 2x + 3\))

The Bottom Line:

This problem tests careful attention to signs and systematic organization of like terms. The key is methodically distributing that negative sign and double-checking arithmetic when combining coefficients.