How do you write the partial fraction decomposition of the rational expression $\frac{x}{(x - 1) (x^{2} + 4)}$ $x/((x-1)(x^2+4)$ ?

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Answer 1 · 2016-11-11T15:42:59

The answer is $= \frac{\frac{1}{5}}{x - 1} + \frac{- \frac{1}{5} x + \frac{4}{5}}{x^{2} + 4}$ $=(1/5)/(x-1)+(-1/5x+4/5)/(x^2+4)$

Let's write the decomposition
$\frac{x}{(x - 1) (x^{2} + 4)} = \frac{A}{x - 1} + \frac{B x + C}{x^{2} + 4}$ $x/((x-1)(x^2+4))=A/(x-1)+(Bx+C)/(x^2+4)$

$= \frac{A (x^{2} + 4) + (B x + C) (x - 1)}{(x - 1) (x^{2} + 4)}$ $=(A(x^2+4)+(Bx+C)(x-1))/((x-1)(x^2+4))$

$:. x=A(x^2+4)+(Bx+C)(x-1)$

Let $x=1$ $=>$ $1=5A$ $=>$ $A=1/5$
Coefficients of $x^2$ $=>$ $0=A+B$ $=>$ $B=-1/5$
And $0=4A-C$ $=>$ $C=4/5$

So, $x/((x-1)(x^2+4))=(1/5)/(x-1)+(-1/5x+4/5)/(x^2+4)$

How do you write the partial fraction decomposition of the rational expression x/((x-1)(x^2+4)x(x−1)(x2+4)x/((x-1)(x^2+4)?