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

Question

1485 views around the world

You can reuse this answer
Creative Commons License

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

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