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)#