Which of the following expressions is equivalent to 24a^3b^2 - 18a^2b^3 + 30a^2b^2?\(2\mathrm{a}^2\mathrm{b}^2(8\mathrm{a} - 6\mathrm{b} + 10)\)\(6\ma...

1. INFER the solution strategy

• Given expression: \(24\mathrm{a}^3\mathrm{b}^2 - 18\mathrm{a}^2\mathrm{b}^3 + 30\mathrm{a}^2\mathrm{b}^2\)
• Strategy: Factor out the greatest common factor (GCF) to create an equivalent expression
• This means finding the largest factor that divides ALL terms

2. SIMPLIFY to find the GCF of coefficients

• Coefficients: \(24, 18, 30\)
• Find prime factorizations: \(24 = 2^3 \times 3\), \(18 = 2 \times 3^2\), \(30 = 2 \times 3 \times 5\)
• GCF = \(2 \times 3 = 6\)

3. SIMPLIFY to find the GCF of variables

• For variable a: Look at powers \(\mathrm{a}^3, \mathrm{a}^2, \mathrm{a}^2\) → take the lowest power = \(\mathrm{a}^2\)
• For variable b: Look at powers \(\mathrm{b}^2, \mathrm{b}^3, \mathrm{b}^2\) → take the lowest power = \(\mathrm{b}^2\)
• Variable GCF = \(\mathrm{a}^2\mathrm{b}^2\)

4. SIMPLIFY by factoring out the complete GCF

• Complete GCF = \(6\mathrm{a}^2\mathrm{b}^2\)
• Divide each term by \(6\mathrm{a}^2\mathrm{b}^2\):
  • \(24\mathrm{a}^3\mathrm{b}^2 \div 6\mathrm{a}^2\mathrm{b}^2 = 4\mathrm{a}\) (since \(24\div6 = 4\) and \(\mathrm{a}^3\div\mathrm{a}^2 = \mathrm{a}^1 = \mathrm{a}\))
  • \((-18\mathrm{a}^2\mathrm{b}^3) \div 6\mathrm{a}^2\mathrm{b}^2 = -3\mathrm{b}\) (since \(-18\div6 = -3\) and \(\mathrm{b}^3\div\mathrm{b}^2 = \mathrm{b}^1 = \mathrm{b}\))
  • \(30\mathrm{a}^2\mathrm{b}^2 \div 6\mathrm{a}^2\mathrm{b}^2 = 5\) (since \(30\div6 = 5\) and \(\mathrm{a}^2\div\mathrm{a}^2 = 1\), \(\mathrm{b}^2\div\mathrm{b}^2 = 1\))

Answer: C. \(6\mathrm{a}^2\mathrm{b}^2(4\mathrm{a} - 3\mathrm{b} + 5)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students incorrectly find the GCF of coefficients, often finding 2 instead of 6 by not systematically using prime factorization or by making arithmetic errors.

When they use GCF = 2 instead of 6, they get:

• \(24\mathrm{a}^3\mathrm{b}^2 \div 2\mathrm{a}^2\mathrm{b}^2 = 12\mathrm{a}\), \((-18\mathrm{a}^2\mathrm{b}^3) \div 2\mathrm{a}^2\mathrm{b}^2 = -9\mathrm{b}\), \(30\mathrm{a}^2\mathrm{b}^2 \div 2\mathrm{a}^2\mathrm{b}^2 = 15\)
• Leading to \(2\mathrm{a}^2\mathrm{b}^2(12\mathrm{a} - 9\mathrm{b} + 15)\), but this doesn't match any choice exactly

However, if they also make variable errors and get \(2\mathrm{a}^2\mathrm{b}^2\), they might get something close to Choice A and select it.

Second Most Common Error:

Weak SIMPLIFY execution with variables: Students take the wrong power when finding the GCF of variables, often taking the highest power instead of the lowest, or making errors with exponent division rules.

For example, taking GCF = \(6\mathrm{ab}\) (using \(\mathrm{a}^1\) and \(\mathrm{b}^1\) instead of \(\mathrm{a}^2\) and \(\mathrm{b}^2\)) leads them toward Choice B: \(6\mathrm{ab}(4\mathrm{a}^2\mathrm{b} - 3\mathrm{ab}^2 + 5\mathrm{a})\).

The Bottom Line:

This problem requires systematic application of GCF principles. Students who rush through finding the GCF or who aren't solid on exponent rules when dividing variables will struggle to get the correct factored form.