Find two rational functions f(x) and g(x) such that f(x) * g(x) = x^(2) and (f(x))/(g(x)) = ((x-1)^2)/((x+2)^2)

Question

Find two rational functions f(x) and g(x) such that
`f(x) * g(x) = x^(2)` and
`(f(x))/(g(x)) = ((x-1)^2)/((x+2)^2)`

Answer 1

`\frac fg=((x-1)^2)/((x+2)^2)`

`\implies f=g((x-1)^2)/((x+2)^2)`

`\implies f^2=fg((x-1)^2)/((x+2)^2)`

`\implies f^2=x^2((x-1)^2)/((x+2)^2)`

`\implies√(f^2)=\sqrt{x^2((x-1)^2)/((x+2)^2)}`

`\implies|f|=\left|(x(x-1))/(x+2)\right|`

Similarly, we can show that


`|g|=\left|(x(x+2))/(x-1)\right|`

Then from here we can solve for
`f` or
`g` by taking either to be both positive, or both negative. That is,


`f(x)=(x(x-1))/(x+2),\,g(x)=(x(x+2))/(x-1)`

or


`f(x)=-(x(x-1))/(x+2),\,g(x)=-(x(x+2))/(x-1)`

(Note that this is only true for
`x\\eq1`,
`x\\eq-2`, and
`x\\eq0`.)