Questions: Perform the indicated operation. 6/(x^2-9) + 3/(x^2-6x+9) - 2/(x^2-2x-15) Simplify your answer.

Transcript text: Perform the indicated operation. \[ \frac{6}{x^{2}-9}+\frac{3}{x^{2}-6 x+9}-\frac{2}{x^{2}-2 x-15} \] $\frac{6}{x^{2}-9}+\frac{3}{x^{2}-6 x+9}-\frac{2}{x^{2}-2 x-15}=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

To simplify the given expression, we need to find a common denominator for the three fractions. This involves factoring the denominators and determining the least common multiple (LCM). Once the common denominator is found, we rewrite each fraction with this common denominator and perform the addition and subtraction of the numerators. Finally, we simplify the resulting expression if possible.

Step 1: Factor the Denominators

The denominators of the given fractions are factored as follows:

$ x^2 - 9 = (x - 3)(x + 3) $
$ x^2 - 6x + 9 = (x - 3)^2 $
$ x^2 - 2x - 15 = (x - 5)(x + 3) $

Step 2: Determine the Common Denominator

The least common multiple (LCM) of the denominators is: \[ \text{Common Denominator} = (x - 5)(x - 3)^3(x + 3)^2 \]

Step 3: Rewrite Each Fraction with the Common Denominator

Each fraction is rewritten as follows: \[ \frac{6}{x^2 - 9} = \frac{6 \cdot (x - 5)(x - 3)^2(x + 3)}{(x - 5)(x - 3)^3(x + 3)^2} \] \[ \frac{3}{x^2 - 6x + 9} = \frac{3 \cdot (x - 5)(x - 3)(x + 3)^2}{(x - 5)(x - 3)^3(x + 3)^2} \] \[ -\frac{2}{x^2 - 2x - 15} = -\frac{2 \cdot (x - 3)^3(x + 3)}{(x - 5)(x - 3)^3(x + 3)^2} \]

Step 4: Combine the Fractions

Combining the fractions results in: \[ \frac{6(x - 5)(x - 3)^2(x + 3) + 3(x - 5)(x - 3)(x + 3)^2 - 2(x - 3)^3(x + 3)}{(x - 5)(x - 3)^3(x + 3)^2} \]

Step 5: Simplify the Result

After simplifying the numerator, we obtain: \[ \frac{7x^4 - 84x^3 + 378x^2 - 1044x + 1863}{x^2 - 8x + 15} \]