Question

How do you write the partial fraction decomposition of the rational expression `(x^2 + 2) /( x^3 + 3x^2 + 3x + 1)`?

Answer 1

The answer is `=3/(x+1)^3-2/(x+1)^2+1/(x+1)`

Explanation:

The denominator is `(x+1)^3`
`:.(x^2+2)/(x+1)^3=A/(x+1)^3+B/(x+1)^2+C/(x+1)`
`=(A+B(x+1)+C(x+1)^2)/(x+1)^3`

`:.x^2+2=A+B(x+1)+C(x+1)^2`

let `x=-1` `=>` `3=A`

If `x=0`, `2=A+B+C`
coefficients of `x^2` `=>` `1=C`
coefficients of `x` `=>` `0=B+2C` `=>` `B=-2`

So, `(x^2+2)/(x+1)^3=3/(x+1)^3-2/(x+1)^2+1/(x+1)`