How do you express (x^3+x^2+x+2)/(x^4+x^2)x3+x2+x+2x4+x2 in partial fractions?

1 Answer
Binayaka C.
Aug 14, 2017

(x^3+x^2+x+2)/(x^2(x^2+1) )= 1/x+ 2/x^2 -1/(x^2+1)x3+x2+x+2x2(x2+1)=1x+2x2−1x2+1

Explanation:

(x^3+x^2+x+2)/(x^4+x^2) = (x^3+x^2+x+2)/(x^2(x^2+1) x3+x2+x+2x4+x2=x3+x2+x+2x2(x2+1)

Let (x^3+x^2+x+2)/(x^2(x^2+1) )= A/x+ B/x^2 + (Cx+D)/(x^2+1)x3+x2+x+2x2(x2+1)=Ax+Bx2+Cx+Dx2+1

Multiplying both sides by x^2(x^2+1)x2(x2+1) we get

(x^3+x^2+x+2)= A(x^3+x)+B(x^2+1) +(Cx+D)x^2(x3+x2+x+2)=A(x3+x)+B(x2+1)+(Cx+D)x2 or

(x^3+x^2+x+2)= Ax^3+Ax+Bx^2+B +Cx^3+Dx^2(x3+x2+x+2)=Ax3+Ax+Bx2+B+Cx3+Dx2 or

(x^3+x^2+x+2)= x^3(A+C)+x^2(B+D)+Ax+B (x3+x2+x+2)=x3(A+C)+x2(B+D)+Ax+B .

Equating with powers of xx 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]

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