Question:Let \(\mathrm{P(x) = 12x^6 - 9x^5 + 3x^4}\). Which expression is equivalent to \(\mathrm{P(x) \div (3x^4)}\)?4x^2 - 3x + 14x^2...

1. TRANSLATE the division problem

• Given: \(\mathrm{P(x) = 12x^6 - 9x^5 + 3x^4}\) divided by \(\mathrm{3x^4}\)
• What this means: We need to divide each term of the polynomial by \(\mathrm{3x^4}\)

2. SIMPLIFY each term division separately

• First term: \(\mathrm{(12x^6) ÷ (3x^4)}\)
  • Coefficient: \(\mathrm{12 ÷ 3 = 4}\)
  • Variable: \(\mathrm{x^6 ÷ x^4 = x^{(6-4)} = x^2}\)
  • Result: \(\mathrm{4x^2}\)
• Second term: \(\mathrm{(-9x^5) ÷ (3x^4)}\)
  • Coefficient: \(\mathrm{-9 ÷ 3 = -3}\)
  • Variable: \(\mathrm{x^5 ÷ x^4 = x^{(5-4)} = x^1 = x}\)
  • Result: \(\mathrm{-3x}\)
• Third term: \(\mathrm{(3x^4) ÷ (3x^4)}\)
  • Coefficient: \(\mathrm{3 ÷ 3 = 1}\)
  • Variable: \(\mathrm{x^4 ÷ x^4 = x^{(4-4)} = x^0 = 1}\)
  • Result: \(\mathrm{1}\)

3. Combine the results

• \(\mathrm{P(x) ÷ (3x^4) = 4x^2 + (-3x) + 1 = 4x^2 - 3x + 1}\)

Answer: A (\(\mathrm{4x^2 - 3x + 1}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when dividing coefficients or applying the exponent rule incorrectly.

Common mistakes include:

• Coefficient errors: Getting \(\mathrm{-9 ÷ 3 = -4}\) instead of \(\mathrm{-3}\)
• Exponent errors: Writing \(\mathrm{x^6 ÷ x^4 = x^2}\) but then using \(\mathrm{x^3}\) in their work
• Sign errors: Losing track of the negative sign on the middle term

This may lead them to select Choice B (\(\mathrm{4x^2 - 3x + 3}\)) if they get the constant term wrong, or Choice C (\(\mathrm{12x^2 - 9x + 3}\)) if they forget to divide the coefficients.

The Bottom Line:

This problem tests whether students can systematically apply the monomial division rule to each term without making computational errors. Success requires careful attention to both coefficient division and exponent subtraction.