Question

Write the partial fraction decomposition fro the ration expression below

-2x + 2y +3z = 7
x - y = -5
y + 4z = -1

1. (5x - 2)/(x-1)^2
2. (x^(3) + x^(2) + x + 2)/(x^(4) + x^(2))

Answer 1

`(5x-2)/((x-1)^2)=\frac a{x-1}+\frac b{(x-1)^2}`

`5x-2=a(x-1)+b`

Since
`5x-2=5x-5+3=5(x-1)+3`, it follows that
`a=5` and
`b=3`, so


`(5x-2)/((x-1)^2)=\frac5{x-1}+\frac3{(x-1)^2}`


`(x^3+x^2+x+2)/(x^4+x^2)=(x^3+x+x+2)/(x^2(x^2+1))=\frac ax+\frac b{x^2}+(cx+d)/(x^2+1)`

`x^3+x^2+x+2=ax(x^2+1)+b(x^2+1)+(cx+d)x^2`

`x^3+x^2+x+2=(a+c)x^3+(b+d)x^2+ax+b`

When
`x=0`, you find that
`b=2`. It's also clear that
`a=1`. So the remaining constants must be
`1+c=1\implies c=0` and
`2+d=1\implies d=-1`, and so you get


`(x^3+x^2+x+2)/(x^4+x^2)=\frac1x+\frac2{x^2}-\frac1{x^2+1}`