\((\mathrm{ax} - 1)(3\mathrm{x}^2 + \mathrm{bx} - 5) = 12\mathrm{x}^3 - \mathrm{cx}^2 - 23\mathrm{x} + 5\) The equation above is true...

1. INFER the solution strategy

Since the equation is true for all x, the polynomials on both sides must be identical. This means I need to expand the left side and match coefficients.

2. SIMPLIFY by expanding the left side

• Expand \((\mathrm{ax} - 1)(3\mathrm{x}^2 + \mathrm{bx} - 5)\):
  • First: \(\mathrm{ax}(3\mathrm{x}^2 + \mathrm{bx} - 5) = 3\mathrm{ax}^3 + \mathrm{abx}^2 - 5\mathrm{ax}\)
  • Second: \(-1(3\mathrm{x}^2 + \mathrm{bx} - 5) = -3\mathrm{x}^2 - \mathrm{bx} + 5\)
  • Combined: \(3\mathrm{ax}^3 + \mathrm{abx}^2 - 5\mathrm{ax} - 3\mathrm{x}^2 - \mathrm{bx} + 5\)

3. SIMPLIFY by collecting like terms

• Group by powers of x:
  • \(\mathrm{x}^3\) terms: \(3\mathrm{ax}^3\)
  • \(\mathrm{x}^2\) terms: \(\mathrm{abx}^2 - 3\mathrm{x}^2 = (\mathrm{ab} - 3)\mathrm{x}^2\)
  • \(\mathrm{x}^1\) terms: \(-5\mathrm{ax} - \mathrm{bx} = (-5\mathrm{a} - \mathrm{b})\mathrm{x}\)
  • Constant: \(5\)

• Result: \(3\mathrm{ax}^3 + (\mathrm{ab} - 3)\mathrm{x}^2 + (-5\mathrm{a} - \mathrm{b})\mathrm{x} + 5\)

4. INFER the coefficient equations

Setting equal to \(12\mathrm{x}^3 - \mathrm{cx}^2 - 23\mathrm{x} + 5\):

• Coefficient of \(\mathrm{x}^3\): \(3\mathrm{a} = 12\)
• Coefficient of \(\mathrm{x}^2\): \(\mathrm{ab} - 3 = -\mathrm{c}\)
• Coefficient of \(\mathrm{x}^1\): \(-5\mathrm{a} - \mathrm{b} = -23\)
• Constant term: \(5 = 5\) ✓

5. SIMPLIFY by solving the system

• From \(\mathrm{x}^3\) equation: \(3\mathrm{a} = 12\)
  \(\mathrm{a} = 4\)

• From \(\mathrm{x}^1\) equation: \(-5\mathrm{a} - \mathrm{b} = -23\)
  \(-5(4) - \mathrm{b} = -23\)
  \(-20 - \mathrm{b} = -23\)
  \(\mathrm{b} = 3\)

• From \(\mathrm{x}^2\) equation: \(\mathrm{ab} - 3 = -\mathrm{c}\)
  \((4)(3) - 3 = -\mathrm{c}\)
  \(12 - 3 = -\mathrm{c}\)
  \(9 = -\mathrm{c}\)
  \(\mathrm{c} = -9\)

Answer: A. -9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{ab} - 3 = -\mathrm{c}\) but forget the negative sign when solving for c.

They calculate \(\mathrm{ab} - 3 = (4)(3) - 3 = 9\), then incorrectly conclude \(\mathrm{c} = 9\) instead of \(\mathrm{c} = -9\). The equation \(\mathrm{ab} - 3 = -\mathrm{c}\) means c is the negative of \((\mathrm{ab} - 3)\).

This leads them to select Choice B (9).

Second Most Common Error:

Poor INFER reasoning: Students get confused about which coefficient corresponds to which term, especially when dealing with the \(-\mathrm{cx}^2\) term.

They might incorrectly associate c with a different coefficient (like the \(\mathrm{x}^3\) coefficient of 12) rather than systematically matching each term.

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically handle polynomial identities and carefully track signs through multi-step algebraic manipulation. The key insight is recognizing that \(\mathrm{ab} - 3 = -\mathrm{c}\) means \(\mathrm{c} = -(\mathrm{ab} - 3)\), not \(\mathrm{c} = \mathrm{ab} - 3\).