Which expression is equivalent to 12x^2y^3 + 18xy^2?

1. TRANSLATE the problem information

• We need to factor: \(12\mathrm{x}^2\mathrm{y}^3 + 18\mathrm{xy}^2\)
• We're looking for an equivalent expression in factored form

2. INFER the approach

• To factor a polynomial, we need to find what's common to both terms
• This means finding the Greatest Common Factor (GCF) of both the coefficients and variables
• Once we find the GCF, we factor it out

3. SIMPLIFY by finding the GCF of coefficients

• \(12 = 2^2 \times 3 = 4 \times 3\)
• \(18 = 2 \times 3^2 = 2 \times 9\)
• The GCF of 12 and 18 is 6

4. SIMPLIFY by finding the GCF of variables

• First term has: \(\mathrm{x}^2\mathrm{y}^3\)
• Second term has: \(\mathrm{xy}^2\)
• For x: we take the smaller power → \(\mathrm{x}^1\)
• For y: we take the smaller power → \(\mathrm{y}^2\)
• Variable GCF = \(\mathrm{xy}^2\)

5. SIMPLIFY by factoring out the complete GCF

• Complete GCF = \(6\mathrm{xy}^2\)
• Divide each term by \(6\mathrm{xy}^2\):
  • \(12\mathrm{x}^2\mathrm{y}^3 \div 6\mathrm{xy}^2 = 2\mathrm{xy}\)
  • \(18\mathrm{xy}^2 \div 6\mathrm{xy}^2 = 3\)
• Result: \(6\mathrm{xy}^2(2\mathrm{xy} + 3)\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make errors when finding the GCF of variables, often taking the larger power instead of the smaller power, or forgetting to include all variable factors.

For example, they might incorrectly identify the variable GCF as \(\mathrm{x}^2\mathrm{y}^3\) instead of \(\mathrm{xy}^2\). This leads to incorrect division and a factored form that doesn't match any of the answer choices, causing confusion and guessing.

Second Most Common Error:

Computational errors in SIMPLIFY: Students correctly identify the GCF as \(6\mathrm{xy}^2\) but make division mistakes when factoring it out.

They might get \(12\mathrm{x}^2\mathrm{y}^3 \div 6\mathrm{xy}^2 = 2\mathrm{x}^2\mathrm{y}\) instead of \(2\mathrm{xy}\) (forgetting to subtract exponents properly). This may lead them to select Choice C (\(6\mathrm{xy}^2(2\mathrm{x}^2\mathrm{y} + 3)\)) which looks similar but has incorrect exponents.

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires careful attention to exponent rules and methodical division - areas where small errors can lead to dramatically different (and incorrect) final expressions.