Which expression is equivalent to 23x^3 + 2x^2 + 9x?

1. INFER the solution approach

• Given expression: \(23\mathrm{x}^3 + 2\mathrm{x}^2 + 9\mathrm{x}\)
• Key insight: Look for common factors among all terms to simplify the expression
• Strategy: Factor out the greatest common factor

2. INFER the greatest common factor

• Examine each term: \(23\mathrm{x}^3, 2\mathrm{x}^2, 9\mathrm{x}\)
• Variable factors: \(\mathrm{x}^3, \mathrm{x}^2, \mathrm{x}\) → greatest common factor is \(\mathrm{x}^1 = \mathrm{x}\)
• Coefficient factors: \(23, 2, 9\) → no common factor (23 is prime, doesn't divide 2 or 9)
• Greatest common factor overall: \(\mathrm{x}\)

3. SIMPLIFY by factoring out x

• Factor x from each term:
  - \(23\mathrm{x}^3 \div \mathrm{x} = 23\mathrm{x}^2\)
  - \(2\mathrm{x}^2 \div \mathrm{x} = 2\mathrm{x}\)
  - \(9\mathrm{x} \div \mathrm{x} = 9\)
• Result: \(23\mathrm{x}^3 + 2\mathrm{x}^2 + 9\mathrm{x} = \mathrm{x}(23\mathrm{x}^2 + 2\mathrm{x} + 9)\)

4. SIMPLIFY verification (optional but recommended)

• Expand to check: \(\mathrm{x}(23\mathrm{x}^2 + 2\mathrm{x} + 9) = 23\mathrm{x}^3 + 2\mathrm{x}^2 + 9\mathrm{x}\) ✓

Answer: C. \(\mathrm{x}(23\mathrm{x}^2 + 2\mathrm{x} + 9)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to factor out a coefficient instead of recognizing that x is the common factor across all terms.

For example, they might notice that 23 appears in the first term and incorrectly try to factor it out of all terms, leading to \(23\mathrm{x}(\mathrm{x}^2 + ?)\), but then get confused when 23 doesn't divide evenly into 2 or 9. This leads to confusion and guessing, or they might select Choice A (\(23\mathrm{x}(\mathrm{x}^2 + 2\mathrm{x} + 9)\)) without properly checking that 23 doesn't factor from all terms.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify x as the common factor but make arithmetic errors when dividing terms.

They might incorrectly compute \(23\mathrm{x}^3 \div \mathrm{x} = 23\mathrm{x}^{3-1} = 23\mathrm{x}^2\) ✓ but then mess up \(2\mathrm{x}^2 \div \mathrm{x} = 2\mathrm{x}^{-1}\) instead of \(2\mathrm{x}^1 = 2\mathrm{x}\), leading to an incorrect factored form that doesn't match any answer choice. This causes them to get stuck and guess.

The Bottom Line:

Success requires systematically identifying the greatest common factor across ALL terms, then carefully performing the division for each term. Students who rush through the factoring process or don't verify their work often select incorrect answers.