If x is a number such that x^3 neq -1, which of the following is equivalent to the expression (x^9...

1. INFER the underlying pattern

Looking at \(\frac{x^9 + 1}{x^3 + 1}\), I need to recognize a key relationship:

• x⁹ can be rewritten as (x³)³
• So the numerator becomes (x³)³ + 1³
• This is a sum of cubes pattern: a³ + b³ where a = x³ and b = 1

2. SIMPLIFY using sum of cubes factorization

• Apply the formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\):
• With \(a = x^3\) and \(b = 1\):
  • \(a + b = x^3 + 1\)
  • \(a^2 - ab + b^2 = (x^3)^2 - (x^3)(1) + (1)^2 = x^6 - x^3 + 1\)
• Therefore: \(x^9 + 1 = (x^3 + 1)(x^6 - x^3 + 1)\)

3. SIMPLIFY the fraction

• Substitute back into the original expression:

\(\frac{x^9 + 1}{x^3 + 1} = \frac{(x^3 + 1)(x^6 - x^3 + 1)}{x^3 + 1}\)

• Cancel the common factor \((x^3 + 1)\):

\(= x^6 - x^3 + 1\)

Answer: B (\(x^6 - x^3 + 1\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(x^9 = (x^3)^3\), so they miss the sum of cubes pattern entirely. Instead, they might try to factor out common terms or attempt polynomial long division, which becomes unnecessarily complex. This often leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students recognize the sum of cubes pattern but make algebraic errors when expanding \((x^3)^2 - (x^3)(1) + 1^2\). Common mistakes include writing \(x^6 + x^3 + 1\) instead of \(x^6 - x^3 + 1\), leading them to select Choice C (\(x^6 + x^3 + 1\)).

The Bottom Line:

This problem tests whether students can see beyond the surface complexity of the ninth power and recognize the elegant sum of cubes structure hiding within. The key insight is that \(x^9\) is really \((x^3)^3\), which transforms a seemingly difficult rational expression into a straightforward application of a fundamental factoring pattern.