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Difficulty: MediumSimplifying and Factoring Algebraic Expressions

For all real numbers xx such that x≠−2x \neq -2, which of the following expressions are equivalent to x6−64x3+8\frac{x^6 - 64}{x^3 + 8}? Select all such expressions.

  1. x3−8x^3 - 8Answer
  2. (x−2)(x2+2x+4)(x - 2)(x^2 + 2x + 4)Answer
  3. C
    (x−2)3(x - 2)^3
  4. D
    x3+8x^3 + 8
  5. E
    x2−8x^2 - 8

Answer

The expressions equivalent to the given rational expression are x3−8x^3 - 8 and (x−2)(x2+2x+4)(x - 2)(x^2 + 2x + 4).
Factoring the numerator x6−64x^6 - 64 as a difference of squares yields (x3−8)(x3+8)(x^3 - 8)(x^3 + 8). Dividing by the denominator (x3+8)(x^3 + 8) leaves x3−8x^3 - 8. Further factoring x3−8x^3 - 8 as a difference of cubes gives (x−2)(x2+2x+4)(x - 2)(x^2 + 2x + 4). Both x3−8x^3 - 8 and (x−2)(x2+2x+4)(x - 2)(x^2 + 2x + 4) are valid equivalent expressions.

Step-by-Step Solution

1
Factor the numerator using the difference of squares identity
x6−64=(x3)2−82=(x3−8)(x3+8)x^6 - 64 = (x^3)^2 - 8^2 = (x^3 - 8)(x^3 + 8)
The expression x6−64x^6 - 64 is a difference of two squares.
2
Simplify the fraction by canceling the common non-zero term (x3+8)(x^3 + 8)
\frac{(x^3 - 8)(x^3 + 8)}{x^3 + 8} = x^3 - 8
Since x≠−2x \neq -2, x3+8≠0x^3 + 8 \neq 0, so (x3+8)(x^3 + 8) can be canceled from numerator and denominator.
3
Factor x3−8x^3 - 8 using the difference of cubes identity
x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)
Applying a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2) with a=xa = x and b=2b = 2.

Key Concept

Simplifying rational expressions by factoring polynomial numerators and denominators using algebraic identities such as difference of squares and difference of cubes.
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Topics

Simplifying and Factoring Algebraic Expressions

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