Which expression is equivalent to 16x^2y^2 + 14xy?

1. TRANSLATE the problem information

• Given expression: \(16\mathrm{x}^2\mathrm{y}^2 + 14\mathrm{xy}\)
• Goal: Find an equivalent factored form from the choices

2. INFER the solution strategy

• This is a factoring problem requiring us to find the Greatest Common Factor (GCF)
• We need to factor out the GCF from both terms
• The GCF will include both numerical and variable components

3. SIMPLIFY to find the GCF

• Coefficients: Find GCF of 16 and 14
  • 16 = \(2^4\), 14 = \(2 \times 7\)
  • GCF = 2
• Variables: Find GCF of \(\mathrm{x}^2\mathrm{y}^2\) and \(\mathrm{xy}\)
  • Take the lowest power of each variable: \(\mathrm{x}^1\) and \(\mathrm{y}^1\)
  • GCF = \(\mathrm{xy}\)
• Complete GCF = \(2\mathrm{xy}\)

4. SIMPLIFY by factoring out the GCF

• Divide each term by \(2\mathrm{xy}\):
  • \(16\mathrm{x}^2\mathrm{y}^2 \div 2\mathrm{xy} = 8\mathrm{xy}\)
  • \(14\mathrm{xy} \div 2\mathrm{xy} = 7\)
• Factored form: \(2\mathrm{xy}(8\mathrm{xy} + 7)\)

5. INFER the correct answer choice

• Compare with options: this matches Choice A exactly

Answer: A. \(2\mathrm{xy}(8\mathrm{xy} + 7)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make errors when dividing polynomial terms by the GCF, especially with exponent subtraction rules.

For example, when computing \(16\mathrm{x}^2\mathrm{y}^2 \div 2\mathrm{xy}\), they might incorrectly get \(8\mathrm{x}^2\mathrm{y}\) instead of \(8\mathrm{xy}\), forgetting that \(\mathrm{x}^2\div\mathrm{x} = \mathrm{x}^1 = \mathrm{x}\) and \(\mathrm{y}^2\div\mathrm{y} = \mathrm{y}^1 = \mathrm{y}\). This type of error would lead them to select Choice B (\(2\mathrm{xy}(8\mathrm{x}^2\mathrm{y} + 7)\)) instead of the correct answer.

Second Most Common Error:

Incomplete INFER reasoning: Students correctly identify that factoring is needed but choose the wrong common factor.

They might factor out \(14\mathrm{xy}\) instead of \(2\mathrm{xy}\) (the actual GCF), leading them to attempt something like \(14\mathrm{xy}(? + 1)\). Since \(16\mathrm{x}^2\mathrm{y}^2\) doesn't divide evenly by \(14\mathrm{xy}\), this creates confusion and may lead them to select Choice C (\(14\mathrm{xy}(2\mathrm{x}^2\mathrm{y} + 1)\)) or give up and guess.

The Bottom Line:

This problem tests whether students can systematically find the GCF of polynomial terms and correctly perform the algebraic division. The key insight is that the GCF must divide evenly into ALL terms, and exponent rules apply when dividing variables.