Simplifying the Expression (x^3  27) / (x  3)
The expression (x^3  27) / (x  3) can be simplified using algebraic manipulation and the concept of factoring. Here's how:
Factoring the Numerator

Recognize the difference of cubes pattern: The numerator, (x^3  27), is a difference of cubes. This pattern can be factored as follows:
 a^3  b^3 = (a  b)(a^2 + ab + b^2)

Apply the pattern: In our case, a = x and b = 3. Therefore:
 x^3  27 = (x  3)(x^2 + 3x + 9)
Simplifying the Expression
Now, our expression becomes: (x^3  27) / (x  3) = [(x  3)(x^2 + 3x + 9)] / (x  3)

Cancel out the common factor: We can cancel out the (x  3) terms in the numerator and denominator, provided x ≠ 3.

Simplified expression: This leaves us with x^2 + 3x + 9.
Conclusion
Therefore, the simplified form of the expression (x^3  27) / (x  3) is x^2 + 3x + 9, where x ≠ 3.