Question

What is the sum?

StartFraction 3 Over x squared minus 9 EndFraction + StartFraction 5 Over x + 3 EndFraction
StartFraction 8 Over x squared + x minus 6 EndFraction
StartFraction 5 x minus 12 Over x minus 3 EndFraction
StartFraction negative 5 x Over (x + 3) (x minus 3) EndFraction
StartFraction 5 x minus 12 Over (x + 3) (x minus 3)

Answer 1

Interpreting the terms as
3/(x^2−9) + 5/(x+3) + 8/(x^2+x−6) + (5x−12)/(x−3) + (−5x)/((x+3)(x−3)) + (5x−12)/((x+3)(x−3)),

note x^2−9 = (x+3)(x−3) and x^2+x−6 = (x+3)(x−2).

First combine the three terms with denominator (x+3)(x−3):
3/(x^2−9) − 5x/((x+3)(x−3)) + (5x−12)/((x+3)(x−3))
= (3 − 5x + 5x − 12)/((x+3)(x−3)) = −9/((x+3)(x−3)).

So the sum becomes
−9/((x+3)(x−3)) + 5/(x+3) + 8/((x+3)(x−2)) + (5x−12)/(x−3).

Putting everything over the common denominator (x+3)(x−3)(x−2) gives the numerator
5x^3 − 2x^2 − 68x + 96.

Thus the simplified sum is
(5x^3 − 2x^2 − 68x + 96)/((x+3)(x−3)(x−2)),

with x ≠ 3, −3, 2.