Which expression is equivalent to 9a^4b^3 + 6a^3b^4 - 12a^2b^5?

1. INFER the solution strategy

• This problem asks for an "equivalent expression," which signals factoring
• Strategy: Factor out the greatest common factor (GCF) from all terms
• We need to find the GCF of coefficients and variables separately, then combine

2. SIMPLIFY to find the GCF of coefficients

• Coefficients: 9, 6, -12
• Find factors: 9 = 3×3, 6 = 3×2, 12 = 3×4
• GCF of coefficients = 3

3. SIMPLIFY to find the GCF of variable terms

• For variable 'a': powers are 4, 3, 2 → GCF = \(\mathrm{a^2}\) (lowest power)
• For variable 'b': powers are 3, 4, 5 → GCF = \(\mathrm{b^3}\) (lowest power)
• Overall variable GCF = \(\mathrm{a^2b^3}\)

4. SIMPLIFY to combine and factor out \(\mathrm{3a^2b^3}\)

• Divide each term by \(\mathrm{3a^2b^3}\):

• \(\mathrm{9a^4b^3 \div 3a^2b^3 = 3a^{(4-2)}b^{(3-3)} = 3a^2}\)
• \(\mathrm{6a^3b^4 \div 3a^2b^3 = 2a^{(3-2)}b^{(4-3)} = 2ab}\)
• \(\mathrm{-12a^2b^5 \div 3a^2b^3 = -4a^{(2-2)}b^{(5-3)} = -4b^2}\)

5. SIMPLIFY to write the final factored form

• \(\mathrm{9a^4b^3 + 6a^3b^4 - 12a^2b^5 = 3a^2b^3(3a^2 + 2ab - 4b^2)}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making exponent subtraction errors when dividing variable terms

Students might incorrectly calculate exponent differences, such as thinking \(\mathrm{a^3 \div a^2 = a}\) instead of \(\mathrm{a^1 = a}\), or miscalculating \(\mathrm{b^5 \div b^3 = b^3}\) instead of \(\mathrm{b^2}\). These computation errors lead to incorrect terms in the factored expression.

This may lead them to select Choice B or get confused between the remaining options.

Second Most Common Error:

Poor INFER reasoning: Not recognizing the systematic approach needed for finding GCF

Students might try to factor by grouping or look for special patterns instead of methodically finding the GCF of coefficients and each variable separately. This leads to incomplete or incorrect factoring attempts.

This causes them to get stuck and guess among the answer choices.

The Bottom Line:

This problem tests whether students can systematically apply the GCF factoring process through multiple algebraic steps without making computational errors along the way.