Partial Fraction Decomposition equating coefficiants

Question

$$\frac{x^2 + x + 1}{(2 x + 1) (x^2 + 1)}$$

I'm having issues with coming with up with the coefficients for this....my conclusion is $1=A+2b \\ 1=2c+b \\ 1=a+c$

am i on the right track? and I'm a little stumped on how to solve the unknowns.

Answer 1

$$ \frac{x^{2}+x+1}{(2x+1)(x^{2}+1)} = \frac{A}{2x+1}+\frac{Bx+C}{x^{2}+1}. $$ Mutliply by $2x+1$ and let $x\rightarrow -1/2$ to obtain $$ \frac{\frac{1}{4}-\frac{1}{2}+1}{\frac{1}{4}+1}=A \implies A=\frac{3}{5}. $$ Set $x=0$: $$ \frac{1}{1}=\frac{A}{1}+\frac{C}{1} \implies 1=A+C \implies C=1-A=\frac{2}{5}. $$ Multiply by $x$ and let $x\rightarrow\infty$ to obtain $$ \frac{1}{2}=\frac{A}{2}+B\implies 1=A+2B\implies B=\frac{C}{2}=\frac{1}{5}. $$