Let \(\mathrm{p(x) = 7x^3 - 4x^2 + 5x - 1}\) and \(\mathrm{q(x) = 3x^3 + 2x^2 - 5x + 8}\)....

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{p(x) = 7x^3 - 4x^2 + 5x - 1}\)
  • \(\mathrm{q(x) = 3x^3 + 2x^2 - 5x + 8}\)
  • \(\mathrm{h(x) = p(x) - q(x)}\)
• Need to find: coefficient of \(\mathrm{x^2}\) in \(\mathrm{h(x)}\)

2. SIMPLIFY the polynomial subtraction

• Set up the subtraction: \(\mathrm{h(x) = (7x^3 - 4x^2 + 5x - 1) - (3x^3 + 2x^2 - 5x + 8)}\)
• Distribute the negative sign carefully:
  \(\mathrm{h(x) = 7x^3 - 4x^2 + 5x - 1 - 3x^3 - 2x^2 + 5x - 8}\)

3. SIMPLIFY by combining like terms

• Group terms by degree:
  • \(\mathrm{x^3}\) terms: \(\mathrm{7x^3 - 3x^3 = 4x^3}\)
  • \(\mathrm{x^2}\) terms: \(\mathrm{-4x^2 - 2x^2 = -6x^2}\)
  • \(\mathrm{x}\) terms: \(\mathrm{5x + 5x = 10x}\)
  • constants: \(\mathrm{-1 - 8 = -9}\)
• Result: \(\mathrm{h(x) = 4x^3 - 6x^2 + 10x - 9}\)

4. Identify the coefficient

• The coefficient of \(\mathrm{x^2}\) in \(\mathrm{h(x)}\) is -6

Answer: C (-6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when distributing the negative sign in polynomial subtraction

Students often write: \(\mathrm{h(x) = 7x^3 - 4x^2 + 5x - 1 - 3x^3 + 2x^2 - 5x + 8}\)

They forget to change the sign of every term in \(\mathrm{q(x)}\), keeping \(\mathrm{+2x^2}\) instead of \(\mathrm{-2x^2}\). This gives them \(\mathrm{-4x^2 + 2x^2 = -2x^2}\) for the \(\mathrm{x^2}\) coefficient.

This may lead them to select Choice B (-2).

Second Most Common Error:

Conceptual confusion about polynomial operations: Attempting to add instead of subtract

Students might misread the problem and compute \(\mathrm{p(x) + q(x)}\) instead of \(\mathrm{p(x) - q(x)}\), giving them \(\mathrm{-4x^2 + 2x^2 = -2x^2}\) or think they should just subtract the absolute values without considering signs properly.

This causes confusion and may lead to selecting Choice B (-2) or guessing among the choices.

The Bottom Line:

This problem tests careful execution of polynomial subtraction, where the critical skill is properly distributing negative signs and accurately combining like terms. The most dangerous step is the sign distribution phase.