In the expression \(3(2\mathrm{x}^2 + \mathrm{px} + 8) - 16\mathrm{x}(\mathrm{p} + 4)\), p is a constant. This expression is equivalent...

1. TRANSLATE the problem information

• Given information:
  • Expression 1: \(\mathrm{3(2x^2 + px + 8) - 16x(p + 4)}\)
  • Expression 2: \(\mathrm{6x^2 - 155x + 24}\)
  • These expressions are equivalent
• What this tells us: Equivalent expressions must have identical coefficients for corresponding terms (x², x, and constant terms)

2. SIMPLIFY the first expression using distributive property

• Expand \(\mathrm{3(2x^2 + px + 8)}\):
  • \(\mathrm{3(2x^2) + 3(px) + 3(8) = 6x^2 + 3px + 24}\)
• Expand \(\mathrm{-16x(p + 4)}\):
  • \(\mathrm{-16x(p) + (-16x)(4) = -16px - 64x}\)
• Combined: \(\mathrm{6x^2 + 3px + 24 - 16px - 64x}\)

3. SIMPLIFY by collecting like terms

• Group by powers of x:
  • x² terms: \(\mathrm{6x^2}\)
  • x terms: \(\mathrm{3px - 16px - 64x = (3p - 16p - 64)x = (-13p - 64)x}\)
  • Constant terms: \(\mathrm{24}\)
• Result: \(\mathrm{6x^2 + (-13p - 64)x + 24}\)

4. INFER the equation from equivalence requirement

• Since \(\mathrm{6x^2 + (-13p - 64)x + 24 ≡ 6x^2 - 155x + 24}\)
• Coefficients must match:
  • x² coefficient: \(\mathrm{6 = 6}\) ✓
  • x coefficient: \(\mathrm{-13p - 64 = -155}\)
  • Constant: \(\mathrm{24 = 24}\) ✓

5. SIMPLIFY to solve for p

• From \(\mathrm{-13p - 64 = -155}\):
  • Add 64 to both sides: \(\mathrm{-13p = -155 + 64}\)
  • \(\mathrm{-13p = -91}\)
  • Divide by -13: \(\mathrm{p = -91/(-13) = 7}\)

Answer: B. 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing negative terms, particularly with \(\mathrm{-16x(p + 4)}\)

They might incorrectly expand this as \(\mathrm{-16xp + 4x}\) instead of \(\mathrm{-16px - 64x}\), leading to wrong coefficients when collecting like terms. This algebraic error propagates through the entire solution, causing them to set up an incorrect equation and solve for the wrong value of p.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate TRANSLATE reasoning: Students don't fully grasp what "equivalent expressions" means mathematically

They might attempt to solve by substituting specific values for x or try to factor both expressions instead of recognizing that equivalent expressions must have identical coefficients for each power of x. This leads them away from the systematic approach needed.

This may lead them to select Choice A (-3) or get stuck and guess randomly.

The Bottom Line:

This problem requires careful algebraic manipulation with attention to signs and systematic coefficient matching. Students who rush through the distributive property or don't understand the meaning of equivalent expressions will struggle to find the correct approach.