Which expression is equivalent to 9c^3 - 4c^3 + c^3?

1. TRANSLATE the problem information

• Given expression: \(9c^3 - 4c^3 + c^3\)
• Need to find: equivalent expression
• Key insight: \(c^3\) is the same as \(1c^3\) (coefficient of 1 is implied)

2. INFER the approach

• All three terms have the same variable part \((c^3)\), making them like terms
• Like terms can be combined by adding/subtracting their coefficients
• Keep the variable part \((c^3)\) unchanged

3. SIMPLIFY by combining coefficients

• Extract the coefficients: \(9, -4, \text{ and } +1\)
• Perform the arithmetic: \(9 - 4 + 1 = 6\)
• Attach the common variable part: \(6c^3\)

Answer: B) \(6c^3\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about exponents: Students incorrectly think they should add the exponents when combining terms.

Instead of combining coefficients \((9 - 4 + 1)\), they add exponents \((3 + 3 + 3 = 9)\), leading to answers with \(c^9\). This may lead them to select Choice C \((6c^9)\) if they get the coefficient calculation right, or Choice D \((14c^9)\) if they make both errors.

Second Most Common Error:

Poor SIMPLIFY execution with signs: Students forget to pay attention to the subtraction sign and incorrectly calculate \(9 + 4 + 1 = 14\) instead of \(9 - 4 + 1 = 6\).

This leads them to select Choice A \((14c^3)\).

The Bottom Line:

This problem tests the fundamental skill of combining like terms - students must recognize that only coefficients are combined while the variable part stays the same, and they must carefully handle positive and negative signs in the arithmetic.