\((\mathrm{ax} + 3)(5\mathrm{x}^2 - \mathrm{bx} + 4) = 20\mathrm{x}^3 - 9\mathrm{x}^2 - 2\mathrm{x} + 12\)The equation above is true for...

1. TRANSLATE the problem information

• Given: \(\mathrm{(ax + 3)(5x^2 - bx + 4) = 20x^3 - 9x^2 - 2x + 12}\)
• The equation is 'true for all x'
• Need to find: \(\mathrm{ab}\)

2. INFER the key insight

• When a polynomial equation is 'true for all x,' it means the polynomials on both sides are identical
• This happens only when coefficients of corresponding terms are exactly equal
• Strategy: Expand the left side, then compare coefficients term by term

3. SIMPLIFY by expanding the left side

• Use distributive property on \(\mathrm{(ax + 3)(5x^2 - bx + 4)}\):
  • \(\mathrm{ax(5x^2 - bx + 4) = 5ax^3 - abx^2 + 4ax}\)
  • \(\mathrm{3(5x^2 - bx + 4) = 15x^2 - 3bx + 12}\)
• Combine: \(\mathrm{5ax^3 - abx^2 + 4ax + 15x^2 - 3bx + 12}\)
• SIMPLIFY by collecting like terms: \(\mathrm{5ax^3 + (-ab + 15)x^2 + (4a - 3b)x + 12}\)

4. INFER coefficient equations

• Left side: \(\mathrm{5ax^3 + (-ab + 15)x^2 + (4a - 3b)x + 12}\)
• Right side: \(\mathrm{20x^3 - 9x^2 - 2x + 12}\)
• For equality: corresponding coefficients must match

5. SIMPLIFY to solve the system

• From \(\mathrm{x^3}\) coefficients: \(\mathrm{5a = 20}\) \(\mathrm{\rightarrow}\) \(\mathrm{a = 4}\)
• From \(\mathrm{x^2}\) coefficients: \(\mathrm{-ab + 15 = -9}\) \(\mathrm{\rightarrow}\) \(\mathrm{-ab = -24}\) \(\mathrm{\rightarrow}\) \(\mathrm{ab = 24}\)

Answer: C. 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing what 'true for all x' means for polynomial equations

Students might try substituting specific values of x (like \(\mathrm{x = 0}\) or \(\mathrm{x = 1}\)) instead of understanding that identical polynomials must have identical coefficients. This approach becomes messy and doesn't lead directly to finding \(\mathrm{ab}\).

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making errors during polynomial expansion

Students correctly understand they need to expand and compare, but make arithmetic mistakes when distributing terms or combining like terms. Common mistakes include:

• Sign errors: \(\mathrm{(-ab + 15)}\) becoming \(\mathrm{(ab + 15)}\)
• Missing terms during distribution
• Calculation errors when solving \(\mathrm{-ab + 15 = -9}\)

This may lead them to select Choice A (18) or Choice B (20) if they get \(\mathrm{a = 4}\) but incorrectly calculate \(\mathrm{b}\).

The Bottom Line:

This problem tests whether students understand that polynomial identity means coefficient-by-coefficient equality, not just numerical equality for specific x-values. The algebraic manipulation is straightforward once this key insight is grasped.