Which of the following expressions is equivalent to 6x^4 + 9x^2 when factored completely?

1. TRANSLATE the problem information

• Given: \(6\mathrm{x}^4 + 9\mathrm{x}^2\)
• Need: Factor completely
• What this means: Find the greatest common factor and factor it out

2. INFER the approach

• To factor completely, I need to find the GCF of all terms
• The GCF will include both coefficient and variable parts
• Factor out this GCF from the entire expression

3. SIMPLIFY to find the GCF of coefficients

• \(6 = 2 \times 3\)
• \(9 = 3^2\)
• GCF of 6 and 9 is 3

4. SIMPLIFY to find the GCF of variables

• First term: \(\mathrm{x}^4\)
• Second term: \(\mathrm{x}^2\)
• GCF is \(\mathrm{x}^2\) (the lowest power of x)

5. INFER the complete GCF

• Coefficient GCF: 3
• Variable GCF: \(\mathrm{x}^2\)
• Complete GCF: \(3\mathrm{x}^2\)

6. SIMPLIFY by factoring out the GCF

• \(6\mathrm{x}^4 + 9\mathrm{x}^2 = 3\mathrm{x}^2(2\mathrm{x}^2 + 3)\)
• Check: \(3\mathrm{x}^2 \times 2\mathrm{x}^2 = 6\mathrm{x}^4\) and \(3\mathrm{x}^2 \times 3 = 9\mathrm{x}^2\) ✓

Answer: B. \(3\mathrm{x}^2(2\mathrm{x}^2 + 3)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Incomplete factoring (INFER skill gap): Students may recognize that choice (C) \(3\mathrm{x}(2\mathrm{x}^3 + 3\mathrm{x})\) expands to the correct expression and select it without checking if it's completely factored.

When they expand \(3\mathrm{x}(2\mathrm{x}^3 + 3\mathrm{x}) = 6\mathrm{x}^4 + 9\mathrm{x}^2\), they think they're done. However, the expression inside the parentheses \((2\mathrm{x}^3 + 3\mathrm{x})\) can be factored further since both terms share a common factor of x. The complete factorization requires factoring out the maximum possible GCF, which is \(3\mathrm{x}^2\), not just \(3\mathrm{x}\).

This may lead them to select Choice C \((3\mathrm{x}(2\mathrm{x}^3 + 3\mathrm{x}))\)

Second Most Common Error:

Incorrect GCF identification (Conceptual gap with exponent rules): Students may incorrectly find the GCF of the variable parts, thinking the GCF of \(\mathrm{x}^4\) and \(\mathrm{x}^2\) is \(\mathrm{x}^4\) instead of \(\mathrm{x}^2\).

This leads them to attempt factoring as \(\mathrm{x}^4(6 + 9/\mathrm{x}^2)\), which doesn't match any answer choice, causing confusion and guessing.

The Bottom Line:

This problem tests whether students understand that "completely factored" means factoring out the greatest possible common factor, not just any common factor that works.