# 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.

## Answer and Explanation:

** 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}**

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from High School Algebra I: Help and Review

Chapter 3 / Lesson 26