Simplify (9-x^2)/(x^3-27)

Problem

(9−x2)/(x3−27)

Solution

1. Factor the numerator by recognizing it as a difference of squares, where 9=3

9−x2=(3−x)*(3+x)

2. Factor the denominator using the difference of cubes formula, a3−b3=(a−b)*(a2+a*b+b2) where a=x and b=3

x3−27=(x−3)*(x2+3*x+9)

3. Rewrite the numerator to create a common factor by factoring out a −1 from (3−x)

(3−x)=−(x−3)

4. Substitute the factored forms back into the original expression.

(−(x−3)*(3+x))/((x−3)*(x2+3*x+9))

5. Cancel the common factor (x−3) from the numerator and the denominator.

(−(3+x))/(x2+3*x+9)

Final Answer

(9−x2)/(x3−27)=−(x+3)/(x2+3*x+9)

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