The partial fraction decomposition of 36/x2 -4 can be written in the form of f(x)/x-2 + g(x)/x+2 , where f(x)= g(x)=

Question

The partial fraction decomposition of 36/x² -4 can be written in the form of f(x)/x-2 + g(x)/x+2 , where

f(x)=

g(x)=

Answer 1

f(x) = 9

g(x) =-9

9/(x-2) + (-9)/(x+2) = 36/(x^2-4)

Answer 2

Let f(x) = A and g(x) = B

36/(x²-4) = A/(x-2) + B/(x+2)

(A(x+2) + B(x-2))/(x²-4) = 36/(x²-4)

Ax + 2A + Bx -2B = 36

x: A+B = 0

36: 2A-2B = 36

A = -B

-2B -2B = 36 --> B = -36/4 = 9 --> A= 9

f(x) = 9

g(x) = -9

You can also use the cover up method since both denominators are linear

36/((x-2)(x+2)) = A/(x-2) + B/(x+2)

To find A, let x = 2 in 36/(x+2) = 36/4 = 9

To find B, let x=-2 in 36/(x-2) = 36/-4 = -9