How do you express #(x^3+x^2+x+2)/(x^4+x^2)# in partial fractions?

1 Answer
Aug 14, 2017

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

Explanation:

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

Let # (x^3+x^2+x+2)/(x^2(x^2+1) )= A/x+ B/x^2 + (Cx+D)/(x^2+1)#

Multiplying both sides by # x^2(x^2+1)# we get

#(x^3+x^2+x+2)= A(x^3+x)+B(x^2+1) +(Cx+D)x^2# or

#(x^3+x^2+x+2)= Ax^3+Ax+Bx^2+B +Cx^3+Dx^2# or

#(x^3+x^2+x+2)= x^3(A+C)+x^2(B+D)+Ax+B # .

Equating with powers of #x# and constant term we get

# B=2 , A=1 , B+D=1, A+C=1 :. C= 0 , D = -1 :.#

# (x^3+x^2+x+2)/(x^2(x^2+1) )= 1/x+ 2/x^2 -1/(x^2+1)# [Ans]