部分分式分解计算器

Your input: perform the partial fraction decomposition of $$$\frac{1}{x \left(x - 48\right)}$$$

The form of the partial fraction decomposition is

$$\frac{1}{x \left(x - 48\right)}=\frac{A}{x}+\frac{B}{x - 48}$$

Write the right-hand side as a single fraction:

$$\frac{1}{x \left(x - 48\right)}=\frac{x B + \left(x - 48\right) A}{x \left(x - 48\right)}$$

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

$$1=x B + \left(x - 48\right) A$$

Expand the right-hand side:

$$1=x A + x B - 48 A$$

Collect up the like terms:

$$1=x \left(A + B\right) - 48 A$$

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

$$\begin{cases} A + B = 0\\- 48 A = 1 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=- \frac{1}{48}$$$, $$$B=\frac{1}{48}$$$

Therefore,

$$\frac{1}{x \left(x - 48\right)}=\frac{- \frac{1}{48}}{x}+\frac{\frac{1}{48}}{x - 48}$$

Answer: $$$\frac{1}{x \left(x - 48\right)}=\frac{- \frac{1}{48}}{x}+\frac{\frac{1}{48}}{x - 48}$$$