Let \(\mathrm{P(p) = 3p^3 - 2p^2 + 6p - 1}\) and \(\mathrm{Q(p) = p^3 + 5p - 4}\). Which of...

1. TRANSLATE the problem information

• We need to find \(\mathrm{P(p) - Q(p)}\) where:
  • \(\mathrm{P(p) = 3p^3 - 2p^2 + 6p - 1}\)
  • \(\mathrm{Q(p) = p^3 + 5p - 4}\)
• This means we subtract the entire second polynomial from the first

2. TRANSLATE the subtraction setup

• Write: \(\mathrm{P(p) - Q(p) = (3p^3 - 2p^2 + 6p - 1) - (p^3 + 5p - 4)}\)
• The parentheses are crucial - we're subtracting ALL terms in \(\mathrm{Q(p)}\)

3. SIMPLIFY by distributing the negative sign

• The negative sign in front of the second set of parentheses affects every term inside
• \(\mathrm{(3p^3 - 2p^2 + 6p - 1) - (p^3 + 5p - 4)}\)
• \(\mathrm{= 3p^3 - 2p^2 + 6p - 1 - p^3 - 5p + 4}\)
• Notice: \(\mathrm{p^3}\) becomes \(\mathrm{-p^3}\), \(\mathrm{+5p}\) becomes \(\mathrm{-5p}\), and \(\mathrm{-4}\) becomes \(\mathrm{+4}\)

4. SIMPLIFY by combining like terms

• Group terms with the same powers:
  • \(\mathrm{p^3}\) terms: \(\mathrm{3p^3 - p^3 = 2p^3}\)
  • \(\mathrm{p^2}\) terms: \(\mathrm{-2p^2}\) (only one \(\mathrm{p^2}\) term)
  • \(\mathrm{p}\) terms: \(\mathrm{6p - 5p = p}\)
  • Constant terms: \(\mathrm{-1 + 4 = 3}\)

5. Write the final expression

• \(\mathrm{P(p) - Q(p) = 2p^3 - 2p^2 + p + 3}\)

Answer: (A) \(\mathrm{2p^3 - 2p^2 + p + 3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Not distributing the negative sign to the constant term properly.

Students correctly distribute the negative to \(\mathrm{p^3}\) and \(\mathrm{5p}\), getting \(\mathrm{-p^3}\) and \(\mathrm{-5p}\), but forget that \(\mathrm{-(-4) = +4}\). Instead, they treat the subtraction as just removing the parentheses: \(\mathrm{-1 - 4 = -5}\).

This leads to: \(\mathrm{2p^3 - 2p^2 + p - 5}\), causing them to select Choice (D).

Second Most Common Error:

Weak SIMPLIFY reasoning: Making calculation errors when combining the \(\mathrm{p}\) terms.

Some students might add instead of subtract: \(\mathrm{6p + 5p = 11p}\) (forgetting that \(\mathrm{5p}\) became \(\mathrm{-5p}\) after distributing the negative sign), or miscalculate \(\mathrm{6p - 5p}\) as \(\mathrm{-p}\) instead of \(\mathrm{p}\).

These errors can lead to selecting Choice (B) or Choice (C) respectively.

The Bottom Line:

Polynomial subtraction requires careful attention to sign changes. The key insight is that subtracting a polynomial means subtracting every single term in it - not just dropping parentheses.