Question

How do you simplify `(1)/(x) - (2)/(x^2 + x) + (3)/(x^3 - x^2)`?

Answer 1

The key to this is to make the denominator the same for all 3 fractions. For this, you have to find the least common multiple of the 3 denominators.
First denominator is just `x`.
Second denominator is `x^2+x=x(x+1)`.
Third denominator is `x^3-x^2=x(x^2-x)=x*x(x-1)`.
The least common multiple is `x*x*(x+1)(x-1)=x^4-x^2`.

For first fraction to have the least common multiple as the denominator we multiply both the numerator and the denominator by `x^3-x`.
`(1*(x^3-x))/(x*(x^3-x))=(x^3-x)/(x^4-x^2)`.
For second fraction we multiply both the numerator and the denominator by `x^2-x`.
`(2(x^2-x))/((x^2+x)(x^2-x))=(2x^2-2x)/(x^4-x^2)`.
For third fraction we multiply both the numerator and the denominator by `x+1`.
`(3(x+1))/((x^3-x^2)(x+1))=(3x+3)/(x^4-x^2)`.

Putting all these together:
`(x^3-x)/(x^4-x^2)-(2x^2-2x)/(x^4-x^2)+(3x+3)/(x^4-x^2)=(x^3-2x^2+4x+3)/(x^4-x^2)`