Which expression is equivalent to 9x^2 + 7x^2 + 9x?

1. INFER which terms can be combined

• Look at each term in the expression: \(9\mathrm{x}^2 + 7\mathrm{x}^2 + 9\mathrm{x}\)
• Identify like terms: \(9\mathrm{x}^2\) and \(7\mathrm{x}^2\) both have \(\mathrm{x}^2\)
• Recognize that \(9\mathrm{x}\) has different power (\(\mathrm{x}^1\)) and cannot be combined with \(\mathrm{x}^2\) terms

2. SIMPLIFY by combining the like terms

• Combine the \(\mathrm{x}^2\) terms: \(9\mathrm{x}^2 + 7\mathrm{x}^2 = (9 + 7)\mathrm{x}^2 = 16\mathrm{x}^2\)
• Keep the \(9\mathrm{x}\) term as is since it has no like terms to combine with
• Write the final simplified expression: \(16\mathrm{x}^2 + 9\mathrm{x}\)

Answer: D. \(16\mathrm{x}^2 + 9\mathrm{x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding what makes terms "like terms" - specifically that both the variable AND the power must be the same.

Students might think that since all terms contain x, they can all be combined together. They might add all coefficients: 9 + 7 + 9 = 25, and incorrectly create something like \(25\mathrm{x}^2\) or \(25\mathrm{x}\). This conceptual confusion about like terms leads to random guessing among the incorrect choices.

Second Most Common Error:

Poor attention to detail during SIMPLIFY: Correctly identifying that \(9\mathrm{x}^2\) and \(7\mathrm{x}^2\) can be combined, but making arithmetic errors when adding the coefficients.

For example, a student might incorrectly calculate 9 + 7 = 15 instead of 16, or might accidentally switch the coefficients around. This may lead them to select Choice B (\(9\mathrm{x}^2 + 16\mathrm{x}\)) where they've confused which coefficient goes with which term.

The Bottom Line:

This problem tests the fundamental skill of recognizing like terms - the building block for all algebraic simplification. Students who struggle here need to strengthen their understanding that terms can only be combined when they have identical variable parts, not just the same variable.