Question

How do you write the partial fraction decomposition of the rational expression `(x^2+9)/(x^4-2x^2-8)`?

Answer 1

`-13/(24(x+2))+13/(24(x-2))-7/(6(x^2+2))`

Explanation:

Factor the denominator.

`x^4-2x^2-8=(x^2-4)(x^2+2)=(x+2)(x-2)(x^2+2)`

`(x^2+9)/((x+2)(x-2)(x^2+2))=A/(x+2)+B/(x-2)+(Cx+D)/(x^2+2)`

Find a common denominator of `(x+2)(x-2)(x^2+2)`.

`x^2+9=A(x-2)(x^2+2)+B(x+2)(x^2+2)+(Cx+D)(x^2-4)`

`x^2+9=Ax^3-2Ax^2+2Ax-4A+Bx^3+2Bx^2+2Bx+4B+Cx^3-4Cx+Dx^2-4D`

`x^2+9=x^3(A+B+C)+x^2(-2A+2B+D)+x(2A+2B+4C)+1(-4A+4B-4D)`

From this, write the following system:
`{(A+B+C=0),(-2A+2B+D=1),(2A+2B+4C=0),(-4A+4B-4D=9):}`

Solve the system:
`{(A=-13/24),(B=13/24),(C=0),(D=-7/6):}`

This gives:

`(x^2+9)/(x^4-2x^2-8)=-13/(24(x+2))+13/(24(x-2))-7/(6(x^2+2))`