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

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Answer 1 · 2015-12-07T01:33:28

Decompose into 3 separate terms ...

#A_1/(x-1)+A_2/(x+1)+A_3/(x+1)^2=1/((x-1)(x+1)^2)#

Now, get a common denominator ...

#[A_1(x+1)^2+A_2(x-1)(x+1)+A_3(x-1)]/[(x-1)(x+1)^2]=1/((x-1)(x+1)^2)#

Now, set the numerators equal...

#A_1(x+1)^2+A_2(x-1)(x+1)+A_3(x-1)=1#

Next, match up the common terms ...

#x^2# terms: #A_1+A_2=0#

#x# terms: #2A_1+A_3=0#

constant terms: #A_1-A_2-A_3=1#

Finally, with 3 equations and 3 unknowns, solve for #A_1, A_2, and A_3#

#A_1=1/4#

#A_2=-1/4#

#A_3=-1/2#

ANSWER :

#A_1/[4(x-1)]-A_2/[4(x+1)]+A_3/[2(x+1)^2]#

hope that helped