Partial Fraction Decomposition Calculator

Your input: perform the partial fraction decomposition of $$$\frac{3 x - 4}{x \left(x - 3\right) \left(x - 2\right)}$$$

The form of the partial fraction decomposition is

$$\frac{3 x - 4}{x \left(x - 3\right) \left(x - 2\right)}=\frac{A}{x}+\frac{B}{x - 3}+\frac{C}{x - 2}$$

Write the right-hand side as a single fraction:

$$\frac{3 x - 4}{x \left(x - 3\right) \left(x - 2\right)}=\frac{x \left(x - 3\right) C + x \left(x - 2\right) B + \left(x - 3\right) \left(x - 2\right) A}{x \left(x - 3\right) \left(x - 2\right)}$$

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

$$3 x - 4=x \left(x - 3\right) C + x \left(x - 2\right) B + \left(x - 3\right) \left(x - 2\right) A$$

Expand the right-hand side:

$$3 x - 4=x^{2} A + x^{2} B + x^{2} C - 5 x A - 2 x B - 3 x C + 6 A$$

Collect up the like terms:

$$3 x - 4=x^{2} \left(A + B + C\right) + x \left(- 5 A - 2 B - 3 C\right) + 6 A$$

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

$$\begin{cases} A + B + C = 0\\- 5 A - 2 B - 3 C = 3\\6 A = -4 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=- \frac{2}{3}$$$, $$$B=\frac{5}{3}$$$, $$$C=-1$$$

Therefore,

$$\frac{3 x - 4}{x \left(x - 3\right) \left(x - 2\right)}=\frac{- \frac{2}{3}}{x}+\frac{\frac{5}{3}}{x - 3}+\frac{-1}{x - 2}$$

Answer: $$$\frac{3 x - 4}{x \left(x - 3\right) \left(x - 2\right)}=\frac{- \frac{2}{3}}{x}+\frac{\frac{5}{3}}{x - 3}+\frac{-1}{x - 2}$$$