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

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Answer 1 · 2016-06-22T11:09:24

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

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

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