Which expression is equivalent to 8x^3 - 5x^2?\(\mathrm{x}^2(8\mathrm{x} - 5)\)\(\mathrm{x}^2(8\mathrm{x} + 5)\)\(8\mathrm{x}^2(\mathrm{x} - 5)\)\(\ma...

1. INFER the problem strategy

• Given: \(8\mathrm{x}^3 - 5\mathrm{x}^2\)
• This expression has multiple terms with common factors, so we need to factor out the greatest common factor (GCF)

2. INFER the greatest common factor

• Look at each term's factors:
  • \(8\mathrm{x}^3\) has factors: 8, x, x, x
  • \(5\mathrm{x}^2\) has factors: 5, x, x
• The GCF is \(\mathrm{x}^2\) (the highest power of x that appears in both terms)

3. SIMPLIFY by factoring out \(\mathrm{x}^2\)

• Factor out \(\mathrm{x}^2\): \(8\mathrm{x}^3 - 5\mathrm{x}^2 = \mathrm{x}^2(? - ?)\)
• Divide each original term by \(\mathrm{x}^2\):
  • \(8\mathrm{x}^3 ÷ \mathrm{x}^2 = 8\mathrm{x}\)
  • \(-5\mathrm{x}^2 ÷ \mathrm{x}^2 = -5\)
• Result: \(\mathrm{x}^2(8\mathrm{x} - 5)\)

4. Verify the factoring

• Check: \(\mathrm{x}^2(8\mathrm{x} - 5) = \mathrm{x}^2 · 8\mathrm{x} - \mathrm{x}^2 · 5 = 8\mathrm{x}^3 - 5\mathrm{x}^2\) ✓

Answer: A. \(\mathrm{x}^2(8\mathrm{x} - 5)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to find the greatest common factor, instead trying to factor by grouping or looking for special patterns that don't apply here.

Without a clear factoring strategy, they may randomly select answers or get confused about which terms can be factored together. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students identify \(\mathrm{x}^2\) as the common factor but make errors when dividing terms, such as:

• Getting \(8\mathrm{x}^3 ÷ \mathrm{x}^2 = 8\mathrm{x}^2\) (forgetting exponent subtraction rule)
• Missing the negative sign: writing \(\mathrm{x}^2(8\mathrm{x} + 5)\) instead of \(\mathrm{x}^2(8\mathrm{x} - 5)\)

This may lead them to select Choice B (\(\mathrm{x}^2(8\mathrm{x} + 5)\)) due to sign errors, or Choice C (\(8\mathrm{x}^2(\mathrm{x} - 5)\)) due to factoring the wrong amount.

The Bottom Line:

This problem tests whether students can systematically identify and extract common factors from polynomial expressions. Success requires both recognizing the factoring strategy and executing it accurately with exponent rules.