Question

Find the partial fraction decomposition of each rational expression.

Answer 1

\[\frac{ x ^{2} }{ (x-1)^{2} (x+1)}\]

I got B and D but having problems finding A and C

Answer 2

there should only be 3 constants
\[\frac{ x ^{2} }{ (x-1)^{2} (x+1)}=\frac{ a }{ x-1 }+\frac{ b }{ (x-1)^2 }+\frac{ c }{ x+1 }\]
multiply by \((x-1)^2(x+1)\)
\[x^2=a(x-1)(x+1)+b(x+1)+c(x-1)^2\]
\[x^2=ax^2-a+bx+b+cx^2-2xc+c\]
equate coefficients to get the system
\[a+c=1\]
\[b-2c=0\]
\[-a+b+c=0\]

Answer 3

b=2c

-a+2c+c=0
-a+3c=0
a=3c
3c+c=1
c=1/4
b=2c=1/2
a=3/4