Question

Express in partial fraction x³ - x² - 4 divided by x² - 1

Answer 1

$\frac{x^3-x^2-4}{x^2-1}$

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

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

$x-1+\frac{x-5}{x^2-1}$

so,$\space \frac{x-5}{x^2-1}=\frac{A}{x+1}+\frac{B}{x-1}$

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

$\begin{cases} a &\text A+B=1 \\ c &\text or\space -A+B=-5 \end{cases}$

Solving A and B then

$A=3\space and \space B=-2$

$so, \space \frac{x-5}{x^2-1}=\frac{3}{x+1}+\frac{-2}{x-1}$

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