(x^3-27)/(x-3)

Jasonbradley

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Jun 17, 2024

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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.