Which expression is equivalent to \((9\mathrm{x}^3 + 5\mathrm{x} + 7) + (6\mathrm{x}^3 + 5\mathrm{x}^2 - 5)\)?

1. TRANSLATE the problem information

• Given: Two polynomials to add: \((9\mathrm{x}^3 + 5\mathrm{x} + 7) + (6\mathrm{x}^3 + 5\mathrm{x}^2 - 5)\)
• Need to find: The equivalent expression after addition

2. SIMPLIFY by identifying and grouping like terms

• Like terms have the same variable raised to the same power
• Group the terms:
  • \(\mathrm{x}^3\) terms: \(9\mathrm{x}^3\) and \(6\mathrm{x}^3\)
  • \(\mathrm{x}^2\) terms: \(5\mathrm{x}^2\) (appears in second polynomial only)
  • \(\mathrm{x}\) terms: \(5\mathrm{x}\) (appears in first polynomial only)
  • Constant terms: \(7\) and \(-5\)

3. SIMPLIFY by combining coefficients of like terms

• \(\mathrm{x}^3\) terms: \(9\mathrm{x}^3 + 6\mathrm{x}^3 = 15\mathrm{x}^3\)
• \(\mathrm{x}^2\) terms: \(5\mathrm{x}^2\) (no other \(\mathrm{x}^2\) terms to combine)
• \(\mathrm{x}\) terms: \(5\mathrm{x}\) (no other \(\mathrm{x}\) terms to combine)
• Constants: \(7 + (-5) = 2\)

4. Write the final expression in standard form

• \(15\mathrm{x}^3 + 5\mathrm{x}^2 + 5\mathrm{x} + 2\)

Answer: D. \(15\mathrm{x}^3 + 5\mathrm{x}^2 + 5\mathrm{x} + 2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Confusing the operations with exponents

Students sometimes think they should multiply the exponents when combining like terms, leading them to get \(\mathrm{x}^6\) instead of \(\mathrm{x}^3\) for the first term. When adding \(9\mathrm{x}^3 + 6\mathrm{x}^3\), they might incorrectly think the powers add up \((3 + 3 = 6)\) rather than understanding that the coefficients add while the power stays the same.

This may lead them to select Choice A (\(15\mathrm{x}^6 + 5\mathrm{x}^2 - 5\mathrm{x} - 35\)) or Choice C (\(15\mathrm{x}^6 + 5\mathrm{x}^2 + 5\mathrm{x} + 2\)).

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors with signs and coefficients

Students might correctly identify like terms but make calculation mistakes, such as incorrectly handling the negative sign in \(-5\) or miscalculating \(7 + (-5)\). They might also miss terms entirely or double-count certain coefficients.

This may lead them to select Choice B (\(15\mathrm{x}^3 + 10\mathrm{x}^2 + 2\)) or other incorrect combinations.

The Bottom Line:

The key to success is systematic identification of like terms and careful coefficient arithmetic. Students who rush through the combining step or don't clearly organize their work are most likely to make errors.