Q. 39

Write the partial fraction decomposition of each rational expression.

$\frac{x^{2}}{x^{3} - 4 x^{2} + 5 x - 2}$

Verified

Answer

The partial fraction decomposition is

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

1Step 1. Given Information.

The rational expression is,

\frac{x^{2}}{x^{3} - 4 x^{2} + 5 x - 2}

2Step 2. Partial Fraction Decomposition.

Decompose the rational fraction, we get,

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

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

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

Therefore,

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

$x^{2} = x^{2} (A + C) + x (- 3 A + B - 2 C) + 2 A - 2 B + C$

Equating the coefficients with the like powers to get,

$A + C = 1 - 3 A + B - 2 C = 0 2 A - 2 B + C = 0$

3Step 3. Solving the equations.

we get,

$A = - 3, B = - 1, C = 4$

$\frac{x^{2}}{(x - 2) {(x - 1)}^{2}} = \frac{- 3}{x - 1} + \frac{- 1}{{(x - 1)}^{2}} + \frac{4}{x - 2}$

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