Question

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

Answer 1

`1-(x+2)/(x^2+x+2).`

Explanation:

We note that the given Rational Fun. is Improper , because the Degree of Poly. in `Nr.=2` = the Degree of Poly. in `Dr.`

Also, the Poly. in `Dr.` can not be factorised, i.e., it is Irreducible.

To make it Proper, usually Long Division is preferred. But, we proceed as under :-

The Expression =`x^2/(x^2+x+2)={(x^2+x+2)-(x+2)}/(x^2+x+2)=(x^2+x+2)/(x^2+x+2)-(x+2)/(x^2+x+2)=1-(x+2)/(x^2+x+2).`

The desired expression in partial fraction is `1-(x+2)/(x^2+x+2).`