Which of the following expressions is(are) a factor of 3x^2 + 20x - 63? x - 9 3x - 7...

1. TRANSLATE the problem information

• Given: Quadratic expression \(3\mathrm{x}^2 + 20\mathrm{x} - 63\)
• Need to check: Which of \(\mathrm{x} - 9\) or \(3\mathrm{x} - 7\) are factors
• What this means: If the expression equals (factor) × (something else), then it's truly a factor

2. INFER the factoring strategy

• For \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\) form, find two numbers that:
  • Multiply to \(\mathrm{ac} = 3(-63) = -189\)
  • Add to \(\mathrm{b} = 20\)
• Then use factor by grouping

3. Find the key numbers

• Need factors of 189: \(1×189, 3×63, 9×21, 27×7\)
• Since product must be -189 (negative), one number positive, one negative
• Testing: \(27 + (-7) = 20\) ✓ and \(27 × (-7) = -189\) ✓

4. SIMPLIFY using factor by grouping

• Rewrite middle term: \(3\mathrm{x}^2 + 20\mathrm{x} - 63 = 3\mathrm{x}^2 + 27\mathrm{x} - 7\mathrm{x} - 63\)
• Group terms: \(= (3\mathrm{x}^2 + 27\mathrm{x}) + (-7\mathrm{x} - 63)\)
• Factor each group: \(= 3\mathrm{x}(\mathrm{x} + 9) - 7(\mathrm{x} + 9)\)
• Factor out common binomial: \(= (3\mathrm{x} - 7)(\mathrm{x} + 9)\)

5. INFER which options match the actual factors

• Actual factors: \((3\mathrm{x} - 7)\) and \((\mathrm{x} + 9)\)
• Option I: \(\mathrm{x} - 9 ≠ \mathrm{x} + 9\), so NOT a factor
• Option II: \(3\mathrm{x} - 7\) matches exactly, so IS a factor

Answer: B. II only

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students find the correct factorization \((3\mathrm{x} - 7)(\mathrm{x} + 9)\) but then carelessly match \(\mathrm{x} - 9\) with \(\mathrm{x} + 9\), thinking they're the same because they both involve 9.

They see \(\mathrm{x} + 9\) as a factor and quickly conclude that \(\mathrm{x} - 9\) (option I) must also be correct, leading them to select Choice C (I and II) instead of recognizing that signs matter crucially in algebra.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make computational errors when finding the two key numbers or during the factor by grouping steps, leading to an incorrect factorization altogether.

This causes them to get stuck and guess, or leads them to incorrectly conclude that neither expression is a factor, selecting Choice D (Neither I nor II).

The Bottom Line:

This problem tests whether students can execute systematic factoring AND carefully match their results to the given options. The real trap is in the details - \(\mathrm{x} + 9\) and \(\mathrm{x} - 9\) look similar but are completely different algebraically.