Write the partial fraction decomposition of the rational exp

Write the partial fraction decomposition of the rational expression. 3x^3 + 2x^2/(x^2 + 5)^2

Solution

given 3x^3 +2x^2 / (x^2 +5)^2

take x^2 common from numerator

3x^3 +2x^2 / (x^2 +5)^2 = x^2(3x+2) / (x^2+5)^2

3x^3 +2x^2 / (x^2 +5)^2 can be partial fraction into Ax +B / (x^2+5) + Cx+D /(x^2+5)^2 [ where A,B,C,D are any numbers]

x^2(3x+2) / (x^2+5)^2 = Ax +B / (x^2+5) + Cx+D /(x^2+5)^2

x^2(3x+2) / (X^2+5)^2 = [ (Ax+B)(x^2+5) + Cx+D ] / (x^2+5)^2

x^2(3x+2) =  [ (Ax+B)(x^2+5) + Cx+D ]

3x^3 + 2x^2 = Ax^3+5Ax+Bx^2 +5B +Cx+D

3x^3 +2x^2 = Ax^3 +Bx^2 + x(5A+C)+5B+D

now comparing the coffients

A=3 (x^3 coffient)

B=2 (x^2 coffient)

5A+C=0 (x coffient)

5(3) +C=0

C= -15

5B+D=0 (constant)

5(2)+D=0

D= -10

x^2(3x+2) / (x^2+5)^2 = Ax +B / (x^2+5) + Cx+D /(x^2+5)^2

plug A,B,C,D values

x^2(3x+2) / (x^2+5)^2 = 3x +2 / (x^2+5) + -15x+-10 /(x^2+5)^2

So OPTION C is ANSWER

C--->ANSWER