# How do you write the partial fraction decomposition of the rational expression (-13x+11) / (2x^3 - 2x^2 + x - 1)?

$\frac{- 13 x + 11}{2 {x}^{3} - 2 {x}^{2} + x - 1} = \frac{- \frac{2}{3}}{x - 1} + \frac{\frac{4}{3} x - \frac{35}{3}}{2 {x}^{2} + 1}$

#### Explanation:

From the given $\frac{- 13 x + 11}{2 {x}^{3} - 2 {x}^{2} + x - 1}$, we start by getting all the factors of the denominator

I will assume that we already know factoring, ok?

$2 {x}^{3} - 2 {x}^{2} + x - 1 = \left(x - 1\right) \left(2 {x}^{2} + 1\right)$

This is our denominator for the right sides of the equation

Let us set up the equation and the variables A, B

$\frac{- 13 x + 11}{2 {x}^{3} - 2 {x}^{2} + x - 1} = \frac{A}{x - 1} + \frac{B x + C}{2 {x}^{2} + 1}$

Simplify using the LCD$= \left(x - 1\right) \left(2 {x}^{2} + 1\right)$

$\frac{- 13 x + 11}{2 {x}^{3} - 2 {x}^{2} + x - 1} = \frac{A \left(2 {x}^{2} + 1\right) + \left(B x + C\right) \left(x - 1\right)}{\left(x - 1\right) \left(2 {x}^{2} + 1\right)}$

Expand then simplify

$\frac{- 13 x + 11}{2 {x}^{3} - 2 {x}^{2} + x - 1} = \frac{2 A {x}^{2} + A + B {x}^{2} - B x + C x - C}{\left(x - 1\right) \left(2 {x}^{2} + 1\right)}$

Rearrange from highest to lowest degree the terms in the numerator at the right side of the equation

$\frac{- 13 x + 11}{2 {x}^{3} - 2 {x}^{2} + x - 1} = \frac{2 A {x}^{2} + B {x}^{2} - B x + C x + A - C}{\left(x - 1\right) \left(2 {x}^{2} + 1\right)}$

Let us match the numerical coefficients of the terms of the numerators of the left and right side of the equation

$\frac{0 \cdot {x}^{2} + \left(- 13\right) {x}^{1} + 11 \cdot {x}^{0}}{2 {x}^{3} - 2 {x}^{2} + x - 1} = \frac{\left(2 A + B\right) {x}^{2} + \left(- B + C\right) {x}^{1} + \left(A - C\right) \cdot {x}^{0}}{\left(x - 1\right) \left(2 {x}^{2} + 1\right)}$

The equations are

$2 A + B = 0$
$- B + C = - 13$
$A - C = 11$

Simultaneous solution results to

$A = - \frac{2}{3}$
$B = \frac{4}{3}$
$C = - \frac{35}{3}$

$\frac{- 13 x + 11}{2 {x}^{3} - 2 {x}^{2} + x - 1} = \frac{- \frac{2}{3}}{x - 1} + \frac{\frac{4}{3} x - \frac{35}{3}}{2 {x}^{2} + 1}$