Question

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

Answer 1

`(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]