%2F(x-3).jpeg "(x^3-27)/(x-3)")
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
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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)
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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)
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Cancel out the common factor: We can cancel out the (x - 3) terms in the numerator and denominator, provided x ≠ 3.
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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.