Which expression is equivalent to 6x^2y^2 + 12x^2y^2?

1. TRANSLATE the problem information

• Given: \(6\mathrm{x}^8\mathrm{y}^2 + 12\mathrm{x}^2\mathrm{y}^2\)
• Find: Equivalent expression in factored form

2. INFER the solution approach

• Both terms share common factors
• Strategy: Factor out the greatest common factor (GCF)
• This will give us an expression in the form: GCF(remaining terms)

3. SIMPLIFY by finding the greatest common factor

• Look at coefficients: 6 and 12 → GCF is 6
• Look at variable parts: \(\mathrm{x}^8\mathrm{y}^2\) and \(\mathrm{x}^2\mathrm{y}^2\) → GCF is \(\mathrm{x}^2\mathrm{y}^2\) (using lowest powers)
• Combined GCF: \(6\mathrm{x}^2\mathrm{y}^2\)

4. SIMPLIFY by factoring out the GCF

• \(6\mathrm{x}^8\mathrm{y}^2 = 6\mathrm{x}^2\mathrm{y}^2 \times \mathrm{x}^6\)
• \(12\mathrm{x}^2\mathrm{y}^2 = 6\mathrm{x}^2\mathrm{y}^2 \times 2\)
• Therefore: \(6\mathrm{x}^8\mathrm{y}^2 + 12\mathrm{x}^2\mathrm{y}^2 = 6\mathrm{x}^2\mathrm{y}^2(\mathrm{x}^6 + 2)\)

5. INFER the correct answer choice

• Compare our result \(6\mathrm{x}^2\mathrm{y}^2(\mathrm{x}^6 + 2)\) with the given options
• This matches Choice C exactly

Answer: C. \(6\mathrm{x}^2\mathrm{y}^2(\mathrm{x}^6 + 2)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly identify the greatest common factor, often focusing only on coefficients or only on variables, but not both systematically.

For example, they might factor out just 6: \(6(\mathrm{x}^8\mathrm{y}^2 + 2\mathrm{x}^2\mathrm{y}^2)\), or just \(\mathrm{x}^2\mathrm{y}^2\): \(\mathrm{x}^2\mathrm{y}^2(6\mathrm{x}^6 + 12)\). This incomplete factoring doesn't match any of the answer choices, leading to confusion and guessing.

Second Most Common Error:

Poor exponent arithmetic: Students make mistakes when determining what remains after factoring out the GCF, especially with the exponent rules.

They might incorrectly think \(6\mathrm{x}^8\mathrm{y}^2 \div 6\mathrm{x}^2\mathrm{y}^2 = \mathrm{x}^4\) instead of \(\mathrm{x}^6\), leading them to select Choice D (\(6\mathrm{x}^2\mathrm{y}^2(\mathrm{x}^4 + 2)\)) instead of the correct answer.

The Bottom Line:

Success requires systematically identifying ALL parts of the greatest common factor (both numerical and variable components) and carefully applying exponent rules when factoring.