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

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Answer 1 · 2016-02-01T08:55:31

Explanation is given below

$(x+1)/(x^2(x-2))$

$(x+1)/(x^2(x-2)) = A/x+B/x^2 + C/(x-2)$

$(x+1)/(x^2(x-2)) = (Ax(x-2) + B(x-2)+Cx^2)/(x^2(x-2))$

Equating the numerators

$x+1 = Ax(x-2)+B(x-2)+Cx^2$

To make this easy for us, let us start by taking $x=0$ and plug in the value, we get.

$0+1=A(0)(0-2)+B(0-2)+C(0)^2$
$1=0-2B+0$

$-2B=1$
$B=-1/2$

Note $x=0$ helped us eliminate $A$ and $C$ now let us take $x=2$
$2+1 = A(2)(2-2)+B(2-2)+C(2^2)$
$3=A(2)(0)+B(0)+4C$
$3=0+0+4C$

$4C=3$
$C=3/4$

We have to find $A$ Let us plug in x=1
$1+1=A(1)(1-2)+B(1-2)+C(1^2)$
$2=A(1)(-1)-B+C$

$-A-B+C = 2$
$-A-(-1/2)+3/4=2$

$-A+1/2+3/4=2$

$-A+2/4+3/4=2$
$-A+5/4=2$

$-A=2-5/4$

$-A=(8-5)/4$

$-A=3/4$

$A=-3/4$

$(x+1)/(x^2(x-2)) = -3/(4x)-1/(2x^2) + 3/(4(x-2))$

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