Which expression is equivalent to -{12x^5 + 18x^4 - 6x^3}?\(6\mathrm{x}^3(-2\mathrm{x}^2 + 3\mathrm{x} - 1)\)\(3\mathrm{x}^3(-4\mathrm{x}^2 + 6\mathrm...

1. INFER the problem approach

• Given: \(-12\mathrm{x}^5 + 18\mathrm{x}^4 - 6\mathrm{x}^3\)
• This is asking for an equivalent expression, and I see a polynomial with multiple terms
• Key insight: Since all answer choices show factored forms, I need to factor out the greatest common factor (GCF)

2. SIMPLIFY by finding the GCF of coefficients

• Coefficients are: 12, 18, 6
• Find GCD: \(12 = 2^2 \times 3\), \(18 = 2 \times 3^2\), \(6 = 2 \times 3\)
• \(\mathrm{GCD}(12, 18, 6) = 2 \times 3 = 6\)

3. SIMPLIFY by finding the GCF of variable terms

• Variable terms are: \(\mathrm{x}^5\), \(\mathrm{x}^4\), \(\mathrm{x}^3\)
• The lowest power of x is \(\mathrm{x}^3\)
• So GCD of variables = \(\mathrm{x}^3\)

4. SIMPLIFY by factoring out the complete GCF

• Overall GCF = \(6\mathrm{x}^3\)
• Factor each term:
  \(-12\mathrm{x}^5 \div 6\mathrm{x}^3 = -2\mathrm{x}^2\)
  \(18\mathrm{x}^4 \div 6\mathrm{x}^3 = 3\mathrm{x}\)
  \(-6\mathrm{x}^3 \div 6\mathrm{x}^3 = -1\)
• Result: \(6\mathrm{x}^3(-2\mathrm{x}^2 + 3\mathrm{x} - 1)\)

5. SIMPLIFY by verifying the answer

• Expand: \(6\mathrm{x}^3(-2\mathrm{x}^2 + 3\mathrm{x} - 1) = -12\mathrm{x}^5 + 18\mathrm{x}^4 - 6\mathrm{x}^3\) ✓
• This matches the original expression

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when factoring out the GCF, particularly with the negative terms.

Students might incorrectly calculate \(-6\mathrm{x}^3 \div 6\mathrm{x}^3 = +1\) instead of \(-1\), or mishandle the negative in \(-12\mathrm{x}^5\). This leads them to get \(6\mathrm{x}^3(-2\mathrm{x}^2 + 3\mathrm{x} + 1)\) instead of the correct \(6\mathrm{x}^3(-2\mathrm{x}^2 + 3\mathrm{x} - 1)\).

This may lead them to select Choice D \((-6\mathrm{x}^3(2\mathrm{x}^2 - 3\mathrm{x} - 1))\) since it contains the wrong signs, or causes confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize they need to find the complete GCF and instead factor out only a partial common factor.

For example, they might factor out just \(3\mathrm{x}^3\) instead of \(6\mathrm{x}^3\), leading to \(3\mathrm{x}^3(-4\mathrm{x}^2 + 6\mathrm{x} - 2)\) which isn't among the choices. Or they factor out \(6\mathrm{x}^2\) instead of \(6\mathrm{x}^3\), missing that all terms contain at least \(\mathrm{x}^3\).

This may lead them to select Choice B \((3\mathrm{x}^3(-4\mathrm{x}^2 + 6\mathrm{x} - 3))\) or Choice C \((6\mathrm{x}^2(-2\mathrm{x}^3 + 3\mathrm{x}^2 - 2\mathrm{x}))\).

The Bottom Line:

This problem tests systematic factoring technique and careful attention to signs. Success requires methodically finding the complete GCF and executing the division accurately, especially with negative terms.