Which expression is equivalent to 8x^3y^2 + 12x^2y^3?

1. INFER what the problem is asking

• We need to find an equivalent expression, which means factoring the given polynomial
• The strategy is to factor out the greatest common factor (GCF) from both terms

2. SIMPLIFY by finding the GCF of the coefficients

• Coefficients: 8 and 12
• Find factors: \(\mathrm{8 = 2^3}\), \(\mathrm{12 = 2^2 \times 3}\)
• GCF of coefficients = 4

3. SIMPLIFY by finding the GCF of the variables

• For x variables: \(\mathrm{x^3}\) and \(\mathrm{x^2}\) → \(\mathrm{GCF = x^2}\) (take the lowest power)
• For y variables: \(\mathrm{y^2}\) and \(\mathrm{y^3}\) → \(\mathrm{GCF = y^2}\) (take the lowest power)
• Overall variable GCF = \(\mathrm{x^2y^2}\)

4. SIMPLIFY by combining the GCF parts

• Complete GCF = \(\mathrm{4x^2y^2}\)

5. SIMPLIFY by factoring out the GCF

• Divide each term by \(\mathrm{4x^2y^2}\):
  • \(\mathrm{8x^3y^2 \div 4x^2y^2 = 2x}\)
  • \(\mathrm{12x^2y^3 \div 4x^2y^2 = 3y}\)
• Write the factored form: \(\mathrm{4x^2y^2(2x + 3y)}\)

Answer: C) \(\mathrm{4x^2y^2(2x + 3y)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly calculate the GCF of the coefficients, often choosing 2 instead of 4, or make errors with the variable powers.

For example, they might think the GCF of 8 and 12 is 2 (instead of 4), leading to factoring out \(\mathrm{2x^2y^2}\) instead of \(\mathrm{4x^2y^2}\). This would give them \(\mathrm{2x^2y^2(4x + 6y)}\), which when they try to simplify further or check against the choices, doesn't match any option cleanly. This leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion about exponent rules: Students forget to use the lowest power when finding the GCF of variables, sometimes taking the highest power instead.

They might incorrectly use \(\mathrm{x^3}\) (highest power) instead of \(\mathrm{x^2}\) (lowest power) for the x-variable GCF. This leads them to attempt factoring with incorrect terms and produces expressions that don't match the answer choices. This may lead them to select Choice A (\(\mathrm{2x^2y^2(4x + 3y)}\)) if they also made coefficient errors.

The Bottom Line:

This problem tests systematic factoring skills - success requires methodical calculation of the GCF for both numerical coefficients and variable terms, combined with careful division to extract the remaining factors.