Question:Let x be a real number with x neq 1 and x neq -1. Which of the following is equivalent...

1. INFER the factorization strategy

• Given: \(\frac{\mathrm{x}^{12} - 1}{\mathrm{x}^2 - 1}\) where \(\mathrm{x} \neq ±1\)
• Key insight: Recognize that \(\mathrm{x}^{12} = (\mathrm{x}^2)^6\), so we can rewrite this as:
  \(\mathrm{x}^{12} - 1 = (\mathrm{x}^2)^6 - 1\)

2. INFER which factorization formula to apply

• Since we have \((\mathrm{x}^2)^6 - 1\), this fits the pattern \(\mathrm{a}^\mathrm{n} - 1\) where \(\mathrm{a} = \mathrm{x}^2\) and \(\mathrm{n} = 6\)
• Apply the formula: \(\mathrm{a}^\mathrm{n} - 1 = (\mathrm{a} - 1)(\mathrm{a}^{\mathrm{n}-1} + \mathrm{a}^{\mathrm{n}-2} + ... + \mathrm{a} + 1)\)
• This gives us: \((\mathrm{x}^2)^6 - 1 = (\mathrm{x}^2 - 1)((\mathrm{x}^2)^5 + (\mathrm{x}^2)^4 + (\mathrm{x}^2)^3 + (\mathrm{x}^2)^2 + (\mathrm{x}^2)^1 + 1)\)

3. SIMPLIFY the factored form

• Expand the powers: \((\mathrm{x}^2)^5 + (\mathrm{x}^2)^4 + (\mathrm{x}^2)^3 + (\mathrm{x}^2)^2 + (\mathrm{x}^2)^1 + 1\)
• This becomes: \(\mathrm{x}^{10} + \mathrm{x}^8 + \mathrm{x}^6 + \mathrm{x}^4 + \mathrm{x}^2 + 1\)

4. SIMPLIFY by canceling common factors

• Our expression is now: \(\frac{(\mathrm{x}^2 - 1)(\mathrm{x}^{10} + \mathrm{x}^8 + \mathrm{x}^6 + \mathrm{x}^4 + \mathrm{x}^2 + 1)}{\mathrm{x}^2 - 1}\)
• Cancel the \((\mathrm{x}^2 - 1)\) terms: \(\mathrm{x}^{10} + \mathrm{x}^8 + \mathrm{x}^6 + \mathrm{x}^4 + \mathrm{x}^2 + 1\)
• Reorder to match answer format: \(1 + \mathrm{x}^2 + \mathrm{x}^4 + \mathrm{x}^6 + \mathrm{x}^8 + \mathrm{x}^{10}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often don't recognize the connection between \(\mathrm{x}^{12} - 1\) and \((\mathrm{x}^2)^6 - 1\), missing the key factorization opportunity.

Instead, they might try to factor \(\mathrm{x}^{12} - 1\) as \((\mathrm{x}^6 - 1)(\mathrm{x}^6 + 1)\) or attempt polynomial long division without recognizing the pattern. This leads to much more complex algebra and often causes them to abandon the systematic approach and guess randomly.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students may correctly identify the factorization approach but make errors in expanding \((\mathrm{x}^2)^5 + (\mathrm{x}^2)^4 + ... + 1\), potentially writing it as the sum of all consecutive powers from \(\mathrm{x}^0\) to \(\mathrm{x}^{11}\).

This may lead them to select Choice A (\(1 + \mathrm{x} + \mathrm{x}^2 + \mathrm{x}^3 + \mathrm{x}^4 + \mathrm{x}^5 + \mathrm{x}^6 + \mathrm{x}^7 + \mathrm{x}^8 + \mathrm{x}^9 + \mathrm{x}^{10} + \mathrm{x}^{11}\)).

The Bottom Line:

This problem tests whether students can recognize patterns in polynomial expressions and apply the correct factorization formulas. The key insight is seeing \(\mathrm{x}^{12} - 1\) as \((\mathrm{x}^2)^6 - 1\), which unlocks a clean factorization that cancels perfectly with the denominator.