How do you write the partial fraction decomposition of the rational expression $(3x^2 + 10x -5)/ ( (x+1)^2(x-2) )$ ?

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Answer 1 · 2016-12-30T09:43:40

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

Let's do the decomposition into partial fractions

$(3x^2+10x-5)/((x+1)^2(x-2))=A/(x+1)^2+B/(x+1)+C/(x-2)$

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

Therefore,

$3x^2+10x-5=A(x-2)+B(x+1)(x-2)+C(x+1)^2$

Let $x=-1$ , $=>$ , $-12=-3A$ , $=>$ , $A=4$

Ler $x=2$ , $=>$ , $27=9C$ , $=>$ , $C=3$

Coefficients of $x^2$

$3=B+C$ , $=>$ , $B=3-C=0$

So,

$(3x^2+10x-5)/((x+1)^2(x-2))=4/(x+1)^2+0/(x+1)+3/(x-2)$

How do you write the partial fraction decomposition of the rational expression (3x^2 + 10x -5)/ ( (x+1)^2(x-2) )(3x^2 + 10x -5)/ ( (x+1)^2(x-2) )?