Partial Fraction Decomposition Calculator

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

Simplify the expression: $$$\frac{1}{y \left(1 - y\right)}=\frac{-1}{y \left(y - 1\right)}$$$

The form of the partial fraction decomposition is

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

Write the right-hand side as a single fraction:

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

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

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

Expand the right-hand side:

$$-1=y A + y B - A$$

Collect up the like terms:

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

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

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

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

Therefore,

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

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