Difference Cubes 901D2B

1. **State the problem:** Simplify or factor the expression $$a^3 - 27(b - c)^3$$. 2. **Recall the formula:** This is a difference of cubes, which follows the formula $$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$$. 3. **Identify terms:** Here, $$x = a$$ and $$y = 3(b - c)$$ because $$27 = 3^3$$. 4. **Apply the formula:** $$a^3 - [3(b - c)]^3 = (a - 3(b - c))\left(a^2 + a \cdot 3(b - c) + [3(b - c)]^2\right)$$ 5. **Simplify inside the parentheses:** $$= (a - 3(b - c))\left(a^2 + 3a(b - c) + 9(b - c)^2\right)$$ 6. **Final factored form:** $$\boxed{(a - 3(b - c))\left(a^2 + 3a(b - c) + 9(b - c)^2\right)}$$ This is the fully factored expression using the difference of cubes formula.