# Use a matrix and Gaussian elimination to solve the PFD equation of \frac{x + 3}{x^2 (x - 1)} =...

## Question:

Use a matrix and Gaussian elimination to solve the PFD equation of

{eq}\frac{x + 3}{x^2 (x - 1)} = \frac{A}{x}+\frac{B}{x^2} +\frac{C}{x - 1}. {/eq}

## Simple Fractions:

The method of decomposition into simple fractions consists of breaking down a ratio of polynomials into a sum of fractions of polynomials of lesser degree. The most important requirement is that the degree of the denominator polynomial is strictly greater than that of the numerator.

Resolve into partial fractions.

{eq}\displaystyle \frac {x+3}{x^2(x-1)}= \frac {A}{x} +\frac {B}{x^2}+\frac {C}{x-1}\\ \displaystyle \frac {x+3}{x^2(x-1)}= \frac {Ax(x-1)+B(x-1)+Cx^2}{ x^2(x-1)}\\ \displaystyle \frac {x+3}{x^2(x-1)}= \frac {Ax^2-Ax+Bx-B+Cx^2}{ x^2(x-1)}\\ \displaystyle \frac {x+3}{x^2(x-1)}= \frac {x^2(A+C)+x(-A+B)-B}{ x^2(x-1)}\\ \displaystyle A+C=0 \\ \displaystyle -A+B=1\\ \displaystyle -B=3 \\ {/eq}

System of equations.

{eq}\begin{bmatrix} C+A+0\cdot B=0 \\ 0\cdot C -A+B=1\\0\cdot C+0\cdot A-B=3\end{bmatrix}\\ {/eq}

The extended matrix

{eq}\begin{bmatrix}1&1&0&|&0\\0&-1&1&|&1\\ 0&0&-1&|&3\end{bmatrix}\\ {/eq}

Calculate the solution of the system of equations

{eq}-B=3 \,\, \Longrightarrow \,\, B=-3\\ -A+B=1 \,\, \Longrightarrow \,\, A=-4\\ A+C=0 \,\, \Longrightarrow \,\, C=4\\ {/eq}

System solution

The system is compatible, it has a unique solution.

{eq}S= \{A=-4, B=-3, C=4 \} {/eq}

The PFD of {eq}\frac {x+3}{x^2(x-1)} {/eq} is:

{eq}\displaystyle \Longrightarrow \boxed{ -\frac {4}{x} -\frac {3}{x^2}+\frac {4}{x-1}}\\ {/eq}