Factor Cube 22B124

1. **State the problem:** Factor the expression $$x^3 - 16$$. 2. **Recognize the form:** This is a difference of cubes because $$16 = 2^4$$ is not a perfect cube, but we can rewrite $$16$$ as $$2^4$$ and check if it fits the difference of cubes formula. However, since $$16$$ is not a perfect cube, we cannot directly apply the difference of cubes formula. 3. **Rewrite the expression:** Notice that $$x^3 - 16$$ can be seen as $$x^3 - 2^4$$. 4. **Try to factor by grouping or other methods:** Since $$16$$ is not a cube, the expression is not a difference of cubes. But we can try to factor it as a difference of squares if possible. 5. **Check for difference of squares:** $$x^3 - 16$$ is not a difference of squares either because $$x^3$$ is not a perfect square. 6. **Use substitution:** Let’s try to factor by expressing $$x^3 - 16$$ as $$x^3 - 2^4$$. Since this is not a standard formula, the expression is prime over the integers. 7. **Conclusion:** The expression $$x^3 - 16$$ cannot be factored further using real numbers and integer coefficients. It is prime. **Final answer:** $$x^3 - 16$$ is prime and cannot be factored further over the integers.