Answer
$\frac{A}{x-1}+\frac{Bx+C}{{{x}^{2}}+1}$.
Work Step by Step
The provided rational expression is as given below:
$\frac{5{{x}^{2}}-6x+7}{\left( x-1 \right)\left( {{x}^{2}}+1 \right)}$
Now, solve the expression as follows:
We set up the partial fraction expansion with unknown constants coefficients and then write a constant coefficients over each of the two distinct algebraic linear factors in the denominator of the expression.
$\frac{5{{x}^{2}}-6x+7}{\left( x-1 \right)\left( {{x}^{2}}+1 \right)}=\frac{A}{x-1}+\frac{Bx+C}{{{x}^{2}}+1}$
Thus, $\frac{A}{x-1}+\frac{Bx+C}{{{x}^{2}}+1}$ is a partial fraction expansion of the rational expression$\frac{5{{x}^{2}}-6x+7}{\left( x-1 \right)\left( {{x}^{2}}+1 \right)}$ with constants $A$, $B$ and $C$ .
