The expression $x^3 - 64$ is a difference of cubes, and its factorization utilizes the difference of cubes formula:

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Steps for $x^3 - 64$:

1. Recognize that $64$ is a cube: $64 = 4^3$.
2. Rewrite $x^3 - 64$ as $x^3 - 4^3$.
3. Apply the formula $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, where $a = x$ and $b = 4$.

$x^3 - 64 = (x - 4)(x^2 + 4x + 16)$

Final Factored Form:

$x^3 - 64 = (x - 4)(x^2 + 4x + 16)$

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Related Questions:

1. How does the sum of cubes formula differ from the difference of cubes formula?
2. What are the roots of the polynomial $x^3 - 64$?
3. How can you verify the factorization of $x^3 - 64$?
4. What is the significance of irreducible quadratic factors in cube factorizations?
5. Can $x^3 - 64$ be further factored over complex numbers?

Tip:

Always check if terms in an expression are perfect cubes before applying the difference or sum of cubes formulas.