Find the partial fraction decomposition of the rational func

Find the partial fraction decomposition of the rational function f(x) given below (do show your work). f(x) = 4x^3 + 3x^2 + 5x - 2/x^3(x + 2)

Solution

f(x)=(4x^3 + 3x^2 + 5x -2)/(x^3)(x+2)

(4x^3+3x^2+5x-2)/x^3(x+2)=A/x + B/x^2 + C/x^3 + D/(x+2)

                                           = (Ax^2(x+2) + Bx(x+2)+ C(x+2) +Dx^3)/(x^3(x+2))

                                        = (Ax^3+ 2Ax^2+ Bx^2+2Bx+Cx+2C+Dx^3)/(x^3(x+2))

   Writing like terms together

                                   (4x^3+3x^2+5x-2)/(x^3(x+2)) = (x^3(A+D) + x^2(2A+B)+x(2B+C) +2C)/(x^3(x+2))

Comparing both sides,we get

A+D=4, 2A+B=3,2B+C=5,2C=-2

Solving them

We get A=0,B=3,C=-1,D=4

Therefore the required partial decomposition is

=0/x + 3/x^2 -1/x^3 + 4/(x+2)

=3/x^2 -1/x^3 + 4/(x+2)