Answer
$ \displaystyle \frac{3x^{3}-5x^{2}+2x+1}{x^{2}(x-1)}$
Work Step by Step
Step 1:
Factor each denominator
The first two are already factored,
$(x-1)^{2},\ x$
$x^{3}-x^{2}=x^{2}(x-1)$
Step 2: The LCM is the product of each of these factors raised to a power equal to the greatest number of times that the factor occurs in the polynomials.
LCM = $x^{2}(x-1)^{2}$
Step 3:
Write each rational expression using the LCM as the denominator. Simplify.
$\displaystyle \frac{x\cdot x^{2}}{(x-1)^{2}\cdot x^{2}}+\frac{2\cdot x(x-1)^{2}}{x\cdot x(x-1)^{2}}-\frac{(x+1)(x-1)}{x^{2}(x-1)(x-1)}$
$ = \displaystyle \frac{x^{3}+2x(x^{2}-2x+1)-(x^{2}-1)}{x^{2}(x-1)^2}$
$ = \displaystyle \frac{x^{3}+2x^{3}-4x^{2}+2x-x^{2}+1}{x^{2}(x-1)^2}$
$= \displaystyle \frac{3x^{3}-5x^{2}+2x+1}{x^{2}(x-1)^2}$
