For nonzero real numbers x and y, which expression is equivalent to (12x^3y - 18x^2y^2 + 6xy)/6xy?2x^2 - 3xy +...

1. TRANSLATE the problem information

• Given: \(\frac{12\mathrm{x}^3\mathrm{y} - 18\mathrm{x}^2\mathrm{y}^2 + 6\mathrm{xy}}{6\mathrm{xy}}\) where \(\mathrm{x} \neq 0, \mathrm{y} \neq 0\)
• Need to find: Equivalent simplified expression

2. INFER the best approach

• We have a polynomial divided by a monomial
• Two strategic options:
  • Factor the numerator first, then cancel
  • Divide each term individually
• Both work equally well - choose what feels more comfortable

3. SIMPLIFY using term-by-term division

• Break apart: \(\frac{12\mathrm{x}^3\mathrm{y} - 18\mathrm{x}^2\mathrm{y}^2 + 6\mathrm{xy}}{6\mathrm{xy}} = \frac{12\mathrm{x}^3\mathrm{y}}{6\mathrm{xy}} - \frac{18\mathrm{x}^2\mathrm{y}^2}{6\mathrm{xy}} + \frac{6\mathrm{xy}}{6\mathrm{xy}}\)

First term:

\(\frac{12\mathrm{x}^3\mathrm{y}}{6\mathrm{xy}} = 2\mathrm{x}^2\)

(12 ÷ 6 = 2, x³ ÷ x = x², y ÷ y = 1)

Second term:

\(\frac{-18\mathrm{x}^2\mathrm{y}^2}{6\mathrm{xy}} = -3\mathrm{xy}\)

(-18 ÷ 6 = -3, x² ÷ x = x, y² ÷ y = y)

Third term:

\(\frac{6\mathrm{xy}}{6\mathrm{xy}} = 1\)

4. SIMPLIFY to final form

• Combine: \(2\mathrm{x}^2 - 3\mathrm{xy} + 1\)

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

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make errors with exponent rules when dividing variables

Many students incorrectly calculate terms like \(\frac{\mathrm{x}^3}{\mathrm{x}}\) or \(\frac{\mathrm{y}^2}{\mathrm{y}}\). For example:

• Thinking \(\frac{\mathrm{x}^3}{\mathrm{x}} = \mathrm{x}^3\) (forgetting to subtract exponents)
• Or calculating \(\frac{-18\mathrm{x}^2\mathrm{y}^2}{6\mathrm{xy}}\) as \(-3\mathrm{x}^2\mathrm{y}^2\) (not reducing the variables properly)

This leads to wrong expressions that don't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly divide some terms but make arithmetic errors in coefficients

They might get the variables right but calculate:

• 12/6 = 3 instead of 2, or
• -18/6 = -2 instead of -3

This systematic error in coefficients can lead them to select Choice (C) (\(2\mathrm{x}^3 - 3\mathrm{x}^2\mathrm{y} + \mathrm{x}\)) if they also mess up the exponent rules.

The Bottom Line:

Success requires careful attention to both coefficient arithmetic AND exponent rules simultaneously. Students often handle one correctly but not the other.