Which expression is equivalent to 5x^2 - 50xy^2?

1. INFER the solution strategy

• The problem asks for an equivalent expression, and I notice both terms share common factors
• Strategy: Factor out the greatest common factor (GCF) to create an equivalent expression

2. SIMPLIFY by identifying the greatest common factor

• First term: \(\mathrm{5x^2 = 5 \cdot x \cdot x}\)
• Second term: \(\mathrm{50xy^2 = 5 \cdot 10 \cdot x \cdot y^2}\)
• Greatest common factor: \(\mathrm{5x}\) (the largest expression that divides both terms)

3. SIMPLIFY by factoring out the GCF

• Divide each term by \(\mathrm{5x}\):
  • \(\mathrm{5x^2 ÷ 5x = x}\)
  • \(\mathrm{50xy^2 ÷ 5x = 10y^2}\)
• Write the factored form: \(\mathrm{5x(x - 10y^2)}\)

4. Verify by checking against answer choices

• My result \(\mathrm{5x(x - 10y^2)}\) matches Choice A exactly

Answer: A. \(\mathrm{5x(x - 10y^2)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly identify that \(\mathrm{5x}\) should be factored out, but make arithmetic errors when dividing the second term.

When dividing \(\mathrm{50xy^2}\) by \(\mathrm{5x}\), they might forget to divide the coefficient: \(\mathrm{50xy^2 ÷ 5x = 50y^2}\) instead of \(\mathrm{10y^2}\). This leads them to write \(\mathrm{5x(x - 50y^2)}\).

This may lead them to select Choice B (\(\mathrm{5x(x - 50y^2)}\))

Second Most Common Error:

Poor INFER reasoning: Students don't recognize factoring as the appropriate strategy and instead try to manipulate the expression in other ways, or they get confused about what "equivalent expression" means.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically apply the factoring process. The key insight is recognizing that factoring creates equivalent expressions, and the execution challenge is careful arithmetic with coefficients.