# How do you simplify (2)/(x) + (2)/(x-1) - (2)/(x-2)?

May 8, 2018

$\frac{2 {x}^{2} - 8 x + 4}{{x}^{3} - 3 {x}^{2} + 2 x}$

#### Explanation:

When adding rational expressions, it is most often the case that you will need to find common denominators. The reason for this is the algebraic rule:

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$

That is, if you have common denominators, you can add the numerators and place them over the common denominator. What follows is a surefire way to find common denominators via a process of multiplying each term by a "clever $1$".

For each term in the sum, multiply the top and bottom of the term by all of the denominators in each other term. In our example, we have three terms, namely $\frac{2}{x}$, $\frac{2}{x - 1}$ and $\frac{2}{x - 2}$. So we multiply the first term by $\frac{\left(x - 1\right) \left(x - 2\right)}{\left(x - 1\right) \left(x - 2\right)}$, since $\left(x - 1\right)$ and $\left(x - 2\right)$ are the denominators for the other terms.

Similarly, we multiply $\frac{2}{x - 1}$ by $\frac{\left(x\right) \left(x - 2\right)}{\left(x\right) \left(x - 2\right)}$ and we multiply $\frac{2}{x - 2}$ by $\frac{\left(x\right) \left(x - 1\right)}{\left(x\right) \left(x - 1\right)}$. Note that we are multiplying each term by 1, so we are not changing the term's value.

Putting this together gives:

$\frac{2}{x} + \frac{2}{x - 1} - \frac{2}{x - 2}$
$= \frac{\left(2\right) \left(x - 1\right) \left(x - 2\right)}{\left(x\right) \left(x - 1\right) \left(x - 2\right)} + \frac{\left(2\right) \left(x\right) \left(x - 2\right)}{\left(x\right) \left(x - 1\right) \left(x - 2\right)} - \frac{\left(2\right) \left(x\right) \left(x - 1\right)}{\left(x\right) \left(x - 1\right) \left(x - 2\right)}$

We have common denominators, so we can add/subtract the numerators and place them over the common denominator.

$\frac{2 \left(x - 1\right) \left(x - 2\right) + 2 x \left(x - 2\right) - 2 x \left(x - 1\right)}{\left(x\right) \left(x - 1\right) \left(x - 2\right)}$
$= \frac{2 \left({x}^{2} - 3 x + 2\right) + 2 {x}^{2} - 4 x - 2 {x}^{2} + 2 x}{\left(x\right) \left({x}^{2} - 3 x + 2\right)}$
$= \frac{2 {x}^{2} - 6 x + 4 + 2 {x}^{2} - 4 x - 2 {x}^{2} + 2 x}{{x}^{3} - 3 {x}^{2} + 2 x}$
$= \frac{2 {x}^{2} - 8 x + 4}{{x}^{3} - 3 {x}^{2} + 2 x}$

The expression cannot be simplified further, so this is our final answer.