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

Jun 22, 2016

$1 - \frac{x + 2}{{x}^{2} + x + 2} .$

#### Explanation:

We note that the given Rational Fun. is Improper , because the Degree of Poly. in $N r . = 2$ = the Degree of Poly. in $D r .$

Also, the Poly. in $D r .$ 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 =${x}^{2} / \left({x}^{2} + x + 2\right) = \frac{\left({x}^{2} + x + 2\right) - \left(x + 2\right)}{{x}^{2} + x + 2} = \frac{{x}^{2} + x + 2}{{x}^{2} + x + 2} - \frac{x + 2}{{x}^{2} + x + 2} = 1 - \frac{x + 2}{{x}^{2} + x + 2} .$

The desired expression in partial fraction is $1 - \frac{x + 2}{{x}^{2} + x + 2} .$