Partial Fraction Decomposition Calculator

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

The form of the partial fraction decomposition is

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

Write the right-hand side as a single fraction:

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

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

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

Expand the right-hand side:

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

Collect up the like terms:

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

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

$$\begin{cases} A + B = 0\\- 2 B + C = 1\\4 A - 2 C = 2 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=\frac{1}{2}$$$, $$$B=- \frac{1}{2}$$$, $$$C=0$$$

Therefore,

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

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