1.14 More on Example 1.3

Solution. We already factorized $g(x)$ as $(x-1)(x^{2}+2x+5)$.

Now we express ${\frac{3x^{2}-x+6}{g(x)}}={\frac{3x^{2}-x+6}{(x-1)(x^{2}+2x+5)}}={\frac{A}{x-1}}+{\frac{Bx+C}{x^{2}+2x+5}}$ where $A,B,C$ are to be determined. To do this we multiply through by $g(x)$, obtaining:

Now the equality of the two sides implies: $A+B=3$, $2A+C-B=-1$, $5A-C=6$. Adding all three equations together we get $A={1,}$ then $B={2,}$ $C={-1.}$