For all real numbers x with x^2 neq 3, the expression (x^4 - 7x^2 + 12)/(x^2 - 3) simplifies to...

1. INFER the strategy needed

• Given: \(\frac{\mathrm{x}^4 - 7\mathrm{x}^2 + 12}{\mathrm{x}^2 - 3}\) should simplify to \(\mathrm{x}^2 + \mathrm{k}\)
• Key insight: The \(\mathrm{x}^4\) and \(\mathrm{x}^2\) terms in the numerator suggest using substitution
• Strategy: Let \(\mathrm{y} = \mathrm{x}^2\) to convert this into a simpler factoring problem

2. TRANSLATE using substitution

• Let \(\mathrm{y} = \mathrm{x}^2\), so \(\mathrm{x}^4 = \mathrm{y}^2\)
• The numerator becomes: \(\mathrm{y}^2 - 7\mathrm{y} + 12\)
• The denominator becomes: \(\mathrm{y} - 3\)
• Our expression is now: \(\frac{\mathrm{y}^2 - 7\mathrm{y} + 12}{\mathrm{y} - 3}\)

3. SIMPLIFY by factoring the numerator

• Need to factor \(\mathrm{y}^2 - 7\mathrm{y} + 12\)
• Find two numbers that multiply to 12 and add to -7
• Those numbers are -3 and -4: \((-3)(-4) = 12\) and \((-3) + (-4) = -7\)
• So \(\mathrm{y}^2 - 7\mathrm{y} + 12 = (\mathrm{y} - 3)(\mathrm{y} - 4)\)

4. SIMPLIFY by canceling common factors

• Our expression becomes: \(\frac{(\mathrm{y} - 3)(\mathrm{y} - 4)}{\mathrm{y} - 3}\)
• Since \(\mathrm{y} \neq 3\) (which means \(\mathrm{x}^2 \neq 3\)), we can cancel \((\mathrm{y} - 3)\)
• Result: \(\mathrm{y} - 4\)

5. TRANSLATE back to original variable

• Substitute \(\mathrm{y} = \mathrm{x}^2\) back in: \(\mathrm{y} - 4 = \mathrm{x}^2 - 4\)
• Since \(\mathrm{x}^2 - 4 = \mathrm{x}^2 + (-4)\), we have \(\mathrm{k} = -4\)

Answer: B (-4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that substitution would simplify the problem, instead attempting to factor \(\mathrm{x}^4 - 7\mathrm{x}^2 + 12\) directly as a quartic polynomial. Students may try grouping methods or other complex approaches that lead nowhere, becoming overwhelmed by the complexity. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when factoring \(\mathrm{y}^2 - 7\mathrm{y} + 12\), such as finding incorrect factor pairs or setting up the factorization incorrectly (like \((\mathrm{y} + 3)(\mathrm{y} + 4)\) instead of \((\mathrm{y} - 3)(\mathrm{y} - 4)\)). Since the problem depends on getting \((\mathrm{y} - 3)\) in the numerator to cancel with the denominator, this error prevents the cancellation step entirely. This may lead them to select Choice A (-3) if they incorrectly factor and get a result involving -3.

The Bottom Line:

This problem tests whether students recognize when substitution can transform a complex polynomial expression into a simpler one. The key insight is seeing that \(\mathrm{x}^4 - 7\mathrm{x}^2 + 12\) has the pattern of a quadratic in \(\mathrm{x}^2\), not attempting to work with it as a quartic.