Simplify the algebraic expression: (x^3 - 8) / (x^2 - 4) for x != 2 and x != -2.

Answer and explanation

Correct answer: B. (x^2 + 2x + 4) / (x + 2)

  1. Step 1: Factor the numerator using the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
  2. Step 2: x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4).
  3. Step 3: Factor the denominator using the difference of squares: x^2 - 4 = (x - 2)(x + 2).
  4. Step 4: Cancel the common non-zero factor (x - 2): (x^2 + 2x + 4) / (x + 2).
Answer
(x^2 + 2x + 4) / (x + 2).
Why not the others: '(x^2 - 2x + 4) / (x + 2)' uses the wrong sign in the quadratic factor of difference of cubes; '(x - 2) / (x + 2)' fails to factor x^3 - 8 properly; '(x^2 + 4) / (x + 2)' drops the middle linear term.

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