Which expression is equivalent to (9p^4q^2 - 12p^3q^3)/3p^3q^2?

1. TRANSLATE the problem information

• Given expression: \(\frac{9\mathrm{p}^4\mathrm{q}^2 - 12\mathrm{p}^3\mathrm{q}^3}{3\mathrm{p}^3\mathrm{q}^2}\)
• Goal: Find an equivalent simplified expression

2. INFER the approach

• Since we have a fraction with polynomials, we can either:
  • Divide each term separately by the denominator, OR
  • Factor the numerator first, then cancel common factors
• Term-by-term division is often more straightforward for this type

3. SIMPLIFY by dividing each term

• First term: \(\frac{9\mathrm{p}^4\mathrm{q}^2}{3\mathrm{p}^3\mathrm{q}^2}\)
  • Coefficients: \(\frac{9}{3} = 3\)
  • Powers of p: \(\frac{\mathrm{p}^4}{\mathrm{p}^3} = \mathrm{p}^1 = \mathrm{p}\)
  • Powers of q: \(\frac{\mathrm{q}^2}{\mathrm{q}^2} = 1\)
  • Result: \(3\mathrm{p}\)

• Second term: \(\frac{12\mathrm{p}^3\mathrm{q}^3}{3\mathrm{p}^3\mathrm{q}^2}\)
  • Coefficients: \(\frac{12}{3} = 4\)
  • Powers of p: \(\frac{\mathrm{p}^3}{\mathrm{p}^3} = 1\)
  • Powers of q: \(\frac{\mathrm{q}^3}{\mathrm{q}^2} = \mathrm{q}^1 = \mathrm{q}\)
  • Result: \(4\mathrm{q}\)

4. SIMPLIFY the final expression

• Combining results: \(3\mathrm{p} - 4\mathrm{q}\)

Answer: A. \(3\mathrm{p} - 4\mathrm{q}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making mistakes with exponent rules, particularly confusing subtraction with addition when dividing powers.

For example, calculating \(\frac{\mathrm{p}^4}{\mathrm{p}^3} = \mathrm{p}^7\) instead of \(\mathrm{p}^1\), or \(\frac{\mathrm{q}^3}{\mathrm{q}^2} = \mathrm{q}^5\) instead of \(\mathrm{q}^1\). Students might also make arithmetic errors with the coefficients (\(\frac{9}{3}\) or \(\frac{12}{3}\)).

This may lead them to select Choice B (\(3\mathrm{p}^2 - 4\mathrm{q}\)) if they incorrectly calculated \(\frac{\mathrm{p}^4}{\mathrm{p}^3} = \mathrm{p}^2\), or other incorrect combinations.

Second Most Common Error:

Poor INFER reasoning: Attempting to factor incorrectly or getting confused about which approach to use.

Some students try to factor but make errors in identifying the greatest common factor, or they start with one method then switch partway through, leading to computational mistakes.

This causes them to get stuck and guess, or arrive at expressions that don't match any answer choice.

The Bottom Line:

Success requires solid mastery of exponent rules and systematic application. The algebraic steps aren't conceptually difficult, but small errors compound quickly in multi-step simplification problems.