部分分式分解计算器

Your input: perform the partial fraction decomposition of $$$\frac{1}{r^{2} \left(r + 1\right)}$$$

The form of the partial fraction decomposition is

$$\frac{1}{r^{2} \left(r + 1\right)}=\frac{A}{r}+\frac{B}{r^{2}}+\frac{C}{r + 1}$$

Write the right-hand side as a single fraction:

$$\frac{1}{r^{2} \left(r + 1\right)}=\frac{r^{2} C + r \left(r + 1\right) A + \left(r + 1\right) B}{r^{2} \left(r + 1\right)}$$

The denominators are equal, so we require the equality of the numerators:

$$1=r^{2} C + r \left(r + 1\right) A + \left(r + 1\right) B$$

Expand the right-hand side:

$$1=r^{2} A + r^{2} C + r A + r B + B$$

Collect up the like terms:

$$1=r^{2} \left(A + C\right) + r \left(A + B\right) + B$$

The coefficients near the like terms should be equal, so the following system is obtained:

$$\begin{cases} A + C = 0\\A + B = 0\\B = 1 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=-1$$$, $$$B=1$$$, $$$C=1$$$

Therefore,

$$\frac{1}{r^{2} \left(r + 1\right)}=\frac{-1}{r}+\frac{1}{r^{2}}+\frac{1}{r + 1}$$

Answer: $$$\frac{1}{r^{2} \left(r + 1\right)}=\frac{-1}{r}+\frac{1}{r^{2}}+\frac{1}{r + 1}$$$