Υπολογιστής ανάλυσης σε μερικά κλάσματα

Your input: perform the partial fraction decomposition of $$$\frac{1}{\left(x^{2} - 20 x\right)^{2}}$$$

Simplify the expression: $$$\frac{1}{\left(x^{2} - 20 x\right)^{2}}=\frac{1}{x^{2} \left(x - 20\right)^{2}}$$$

The form of the partial fraction decomposition is

$$\frac{1}{x^{2} \left(x - 20\right)^{2}}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x - 20}+\frac{D}{\left(x - 20\right)^{2}}$$

Write the right-hand side as a single fraction:

$$\frac{1}{x^{2} \left(x - 20\right)^{2}}=\frac{x^{2} \left(x - 20\right) C + x^{2} D + x \left(x - 20\right)^{2} A + \left(x - 20\right)^{2} B}{x^{2} \left(x - 20\right)^{2}}$$

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

$$1=x^{2} \left(x - 20\right) C + x^{2} D + x \left(x - 20\right)^{2} A + \left(x - 20\right)^{2} B$$

Expand the right-hand side:

$$1=x^{3} A + x^{3} C - 40 x^{2} A + x^{2} B - 20 x^{2} C + x^{2} D + 400 x A - 40 x B + 400 B$$

Collect up the like terms:

$$1=x^{3} \left(A + C\right) + x^{2} \left(- 40 A + B - 20 C + D\right) + x \left(400 A - 40 B\right) + 400 B$$

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

$$\begin{cases} A + C = 0\\- 40 A + B - 20 C + D = 0\\400 A - 40 B = 0\\400 B = 1 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=\frac{1}{4000}$$$, $$$B=\frac{1}{400}$$$, $$$C=- \frac{1}{4000}$$$, $$$D=\frac{1}{400}$$$

Therefore,

$$\frac{1}{x^{2} \left(x - 20\right)^{2}}=\frac{\frac{1}{4000}}{x}+\frac{\frac{1}{400}}{x^{2}}+\frac{- \frac{1}{4000}}{x - 20}+\frac{\frac{1}{400}}{\left(x - 20\right)^{2}}$$

Answer: $$$\frac{1}{\left(x^{2} - 20 x\right)^{2}}=\frac{\frac{1}{4000}}{x}+\frac{\frac{1}{400}}{x^{2}}+\frac{- \frac{1}{4000}}{x - 20}+\frac{\frac{1}{400}}{\left(x - 20\right)^{2}}$$$