Mathematics

Problem

Express the rational function

G(x)=\frac{x^{3}+2x^{2}+3}{x^{2}+1}

in the form

G(x)=P(x)+\frac{Ax+B}{x^{2}+1},

where

P(x)

is a polynomial and

A,\,B

are real constants. Proceed through the following steps: (i) Perform polynomial division to obtain

P(x)

and the remainder. (ii) Find

A

and

B

so that the remainder term is written as

\frac{Ax+B}{x^{2}+1}.

(iii) Combine the results to give the complete expression for

G(x)