Question:Which expression is equivalent to 6x^2 - 24x?Answer Choices:\(6\mathrm{x}(\mathrm{x} - 4)\)\(6\mathrm{x}(\mathrm{x} - 24)\)\(6\mathrm{x}^2(\m...

1. INFER what the problem is asking

• The question asks for an 'equivalent expression'
• This tells us we need to factor the polynomial \(6\mathrm{x}^2 - 24\mathrm{x}\)
• Strategy: Find the greatest common factor (GCF) and factor it out

2. SIMPLIFY by finding the GCF

• Break down each term into its factors:
  • \(6\mathrm{x}^2 = 6 \times \mathrm{x} \times \mathrm{x}\)
  • \(24\mathrm{x} = 6 \times 4 \times \mathrm{x}\)
• The GCF is \(6\mathrm{x}\) (the common factors that appear in both terms)

3. SIMPLIFY by factoring out the GCF

• Factor out \(6\mathrm{x}\): \(6\mathrm{x}^2 - 24\mathrm{x} = 6\mathrm{x}(\mathrm{x} - 4)\)
• Check: \(6\mathrm{x}(\mathrm{x} - 4) = 6\mathrm{x}^2 - 24\mathrm{x}\) ✓

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Incomplete SIMPLIFY execution: Students find an incomplete GCF, factoring out only part of the common factors.

For example, they might factor out just 6 to get \(6(\mathrm{x}^2 - 4\mathrm{x})\), or just x to get \(\mathrm{x}(6\mathrm{x} - 24)\). Neither of these matches any answer choice, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students incorrectly identify \(6\mathrm{x}^2\) as the GCF instead of \(6\mathrm{x}\).

This leads them to attempt factoring as: \(6\mathrm{x}^2 - 24\mathrm{x} = 6\mathrm{x}^2(\text{something})\). To make this work, they might write \(6\mathrm{x}^2(\mathrm{x} - 4)\) or \(6\mathrm{x}^2(4 - \mathrm{x})\), leading them to select Choice C or Choice D.

The Bottom Line:

Success depends on systematically finding the complete GCF by identifying all common factors (both numerical and variable parts), then carefully executing the factoring steps.