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

1. TRANSLATE the problem information

• Given: \((7\mathrm{x}^3 + 7\mathrm{x}) - (6\mathrm{x}^3 - 3\mathrm{x})\)
• Need to find: equivalent expression from the choices

2. INFER the approach

• The subtraction of the second polynomial requires distributing the negative sign
• After distribution, we'll need to identify and combine like terms
• This will SIMPLIFY to a cleaner expression

3. SIMPLIFY by distributing the negative sign

• \((7\mathrm{x}^3 + 7\mathrm{x}) - (6\mathrm{x}^3 - 3\mathrm{x})\)
• \(= 7\mathrm{x}^3 + 7\mathrm{x} - 6\mathrm{x}^3 - (-3\mathrm{x})\)
• \(= 7\mathrm{x}^3 + 7\mathrm{x} - 6\mathrm{x}^3 + 3\mathrm{x}\)

4. SIMPLIFY by grouping like terms

• Group x³ terms together and x terms together:
• \(= (7\mathrm{x}^3 - 6\mathrm{x}^3) + (7\mathrm{x} + 3\mathrm{x})\)

5. SIMPLIFY by combining like terms

• \(= (1)\mathrm{x}^3 + (10)\mathrm{x}\)
• \(= \mathrm{x}^3 + 10\mathrm{x}\)

Answer: A. \(\mathrm{x}^3 + 10\mathrm{x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing the negative sign over the second polynomial.

They might write: \((7\mathrm{x}^3 + 7\mathrm{x}) - (6\mathrm{x}^3 - 3\mathrm{x}) = 7\mathrm{x}^3 + 7\mathrm{x} - 6\mathrm{x}^3 - 3\mathrm{x}\)

This gives them \((7\mathrm{x}^3 - 6\mathrm{x}^3) + (7\mathrm{x} - 3\mathrm{x}) = \mathrm{x}^3 + 4\mathrm{x}\)

This may lead them to select Choice D (\(\mathrm{x}^3 + 4\mathrm{x}\))

Second Most Common Error:

Poor INFER reasoning: Students might try to combine terms that aren't actually like terms, or add coefficients instead of subtracting them properly.

For example, they might incorrectly think \(7\mathrm{x}^3 - 6\mathrm{x}^3\) gives them something like \(13\mathrm{x}^3\) instead of \(\mathrm{x}^3\), leading to confusion and potentially guessing among the remaining choices.

The Bottom Line:

This problem tests careful execution of the distributive property with negative signs. The key insight is that subtracting a polynomial means distributing that negative sign to every term inside the parentheses - missing this creates cascading errors through the rest of the problem.