If \((\mathrm{x} - 3)(2\mathrm{x} + \mathrm{p}) = 2\mathrm{x}^2 + \mathrm{qx} - 15\), and q = -1, what is the value...

1. SIMPLIFY the left side by expanding

• We need to multiply out \((\mathrm{x} - 3)(2\mathrm{x} + \mathrm{p})\) using the distributive property:
  • First: \(\mathrm{x} \times (2\mathrm{x} + \mathrm{p}) = 2\mathrm{x}^2 + \mathrm{px}\)
  • Then: \(-3 \times (2\mathrm{x} + \mathrm{p}) = -6\mathrm{x} - 3\mathrm{p}\)
  • Combine: \(2\mathrm{x}^2 + \mathrm{px} - 6\mathrm{x} - 3\mathrm{p} = 2\mathrm{x}^2 + (\mathrm{p} - 6)\mathrm{x} - 3\mathrm{p}\)

2. INFER the coefficient matching strategy

• Since we have: \(2\mathrm{x}^2 + (\mathrm{p} - 6)\mathrm{x} - 3\mathrm{p} = 2\mathrm{x}^2 + \mathrm{qx} - 15\)
• For these polynomials to be equal, their coefficients must match term by term
• This gives us two equations to work with

3. SIMPLIFY to find p using the x-coefficient

• Coefficient of x: \((\mathrm{p} - 6) = \mathrm{q} = -1\)
• Solve: \(\mathrm{p} - 6 = -1\)
• Therefore: \(\mathrm{p} = 5\)

4. INFER verification using the constant term

• As a check: constant terms must also match
• We have: \(-3\mathrm{p} = -15\)
• This gives: \(\mathrm{p} = 5\) ✓

Answer: D) 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \((\mathrm{x} - 3)(2\mathrm{x} + \mathrm{p})\), particularly when distributing the -3 or combining like terms. For example, they might get \(2\mathrm{x}^2 + \mathrm{px} - 6\mathrm{x} + 3\mathrm{p}\) instead of \(2\mathrm{x}^2 + \mathrm{px} - 6\mathrm{x} - 3\mathrm{p}\), missing the negative sign on \(3\mathrm{p}\).

This leads to incorrect coefficient equations like \((\mathrm{p} - 6) = -1\) but \(3\mathrm{p} = -15\), giving contradictory values for p. This causes confusion and often leads to guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students correctly expand but don't recognize they need to match coefficients systematically. Instead, they might try to substitute specific x-values or use other inefficient approaches.

This may lead them to select Choice A (-5) if they incorrectly think p = q, or get stuck and guess randomly.

The Bottom Line:

This problem tests whether students can execute polynomial multiplication accurately and recognize the fundamental principle that equal polynomials have equal coefficients. The algebra must be precise, as small errors cascade through the solution.