\((4\mathrm{x}^3 - 5\mathrm{x}^2 + 3) - (6\mathrm{x}^3 + 2\mathrm{x}^2 - \mathrm{x})\) Which of the following expressions is equivalent to the...

1. INFER the problem structure

• We have polynomial subtraction: (first polynomial) - (second polynomial)
• Strategy: Distribute the negative sign first, then combine like terms

2. SIMPLIFY by distributing the negative

• Remove parentheses using distributive property:

\((4x^3 - 5x^2 + 3) - (6x^3 + 2x^2 - x)\)

\(= 4x^3 - 5x^2 + 3 - 6x^3 - 2x^2 + x\)

3. SIMPLIFY by combining like terms

• Group terms with same variables and exponents:

• \(x^3\) terms: \(4x^3 - 6x^3 = -2x^3\)
• \(x^2\) terms: \(-5x^2 - 2x^2 = -7x^2\)
• \(x\) terms: \(+x\) (only one x term)
• constants: \(+3\) (only one constant)

• Final expression: \(-2x^3 - 7x^2 + x + 3\)

Answer: B. \(-2x^3 - 7x^2 + x + 3\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly distribute the negative sign, treating subtraction like addition.

They might write: \((4x^3 - 5x^2 + 3) - (6x^3 + 2x^2 - x) = 4x^3 - 5x^2 + 3 + 6x^3 + 2x^2 + x\)

This leads to: \(10x^3 - 3x^2 + x + 3\), causing them to select Choice A (\(-10x^3 - 3x^2 + x + 3\)) after making a sign error on the \(x^3\) coefficient.

Second Most Common Error:

Poor SIMPLIFY execution: Students distribute correctly but make arithmetic errors when combining like terms, particularly with the \(x^2\) coefficients.

They might calculate: \(-5x^2 - 2x^2 = -3x^2\) instead of \(-7x^2\)

This leads them to select Choice C (\(-2x^3 - 3x^2 + x + 3\)).

The Bottom Line:

Polynomial subtraction requires careful attention to sign changes when distributing and methodical combination of like terms. The negative sign in front of parentheses changes every term inside—this is the critical insight students often miss.