Which expression is equivalent to 8p^2 - 32pq^2?Choose 1 answer:\(8\mathrm{p}(\mathrm{p} - 4\mathrm{q}^2)\)\(8\mathrm{p}(\mathrm{p} - 32\mathrm{q}^2)\...

1. INFER the solution strategy

• Given expression: \(8\mathrm{p}^2 - 32\mathrm{pq}^2\)
• Strategy: Factor this expression by finding the greatest common factor (GCF)
• This approach works because both terms share common factors that can be pulled out

2. INFER the greatest common factor

• Analyze the first term (\(8\mathrm{p}^2\)): Contains factors 8, p, and p
• Analyze the second term (\(32\mathrm{pq}^2\)): Contains factors 8, 4, p, q, and q
• Common factors present in both terms: 8 and p
• Therefore, \(\mathrm{GCF} = 8\mathrm{p}\)

3. SIMPLIFY by factoring out the GCF

• Factor out 8p: \(8\mathrm{p}^2 - 32\mathrm{pq}^2 = 8\mathrm{p}(? - ?)\)
• Find what's left in parentheses by dividing each original term by 8p:
  • \(8\mathrm{p}^2 \div 8\mathrm{p} = \mathrm{p}\)
  • \(32\mathrm{pq}^2 \div 8\mathrm{p} = 4\mathrm{q}^2\)
• Result: \(8\mathrm{p}^2 - 32\mathrm{pq}^2 = 8\mathrm{p}(\mathrm{p} - 4\mathrm{q}^2)\)

4. Verify the factorization

• Expand: \(8\mathrm{p}(\mathrm{p} - 4\mathrm{q}^2) = 8\mathrm{p} \cdot \mathrm{p} - 8\mathrm{p} \cdot 4\mathrm{q}^2 = 8\mathrm{p}^2 - 32\mathrm{pq}^2\) ✓

Answer: A. \(8\mathrm{p}(\mathrm{p} - 4\mathrm{q}^2)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly identify the greatest common factor as just 8 (instead of 8p), or as \(8\mathrm{p}^2\) (instead of 8p).

When they factor out only 8: \(8\mathrm{p}^2 - 32\mathrm{pq}^2 = 8(\mathrm{p}^2 - 4\mathrm{pq}^2)\)
When they factor out \(8\mathrm{p}^2\): They get confused because \(32\mathrm{pq}^2\) doesn't contain \(\mathrm{p}^2\)

This confusion about the GCF may lead them to select Choice C (\(8\mathrm{p}^2(\mathrm{p} - 4\mathrm{q}^2)\)) or causes them to get stuck and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify 8p as the GCF but make division errors when determining what remains in parentheses.

They might incorrectly calculate: \(32\mathrm{pq}^2 \div 8\mathrm{p} = 32\mathrm{q}^2\) (forgetting to divide the coefficient 32 by 8)

This may lead them to select Choice B (\(8\mathrm{p}(\mathrm{p} - 32\mathrm{q}^2)\)).

The Bottom Line:

Success requires both strategic thinking (recognizing the need for GCF) and careful algebraic execution (correctly identifying all common factors and performing accurate division).