# How do you simplify a/(1-a) + (3a)/(a+1 )- 5/(a^2-1)?

Sep 4, 2016

$\frac{a}{1 - a} + \frac{3 a}{a + 1} - \frac{5}{{a}^{2} - 1} = \frac{2 {a}^{2} - 4 a - 5}{{a}^{2} - 1}$

$\textcolor{w h i t e}{\frac{a}{1 - a} + \frac{3 a}{a + 1} - \frac{5}{{a}^{2} - 1}} = 2 - \frac{4 a + 3}{{a}^{2} - 1}$

#### Explanation:

Note that ${a}^{2} - 1 = \left(a - 1\right) \left(a + 1\right)$

So we find:

$\frac{a}{1 - a} + \frac{3 a}{a + 1} - \frac{5}{{a}^{2} - 1} = \frac{- a \left(a + 1\right) + 3 a \left(a - 1\right) - 5}{{a}^{2} - 1}$

$\textcolor{w h i t e}{\frac{a}{1 - a} + \frac{3 a}{a + 1} - \frac{5}{{a}^{2} - 1}} = \frac{- {a}^{2} - a + 3 {a}^{2} - 3 a - 5}{{a}^{2} - 1}$

$\textcolor{w h i t e}{\frac{a}{1 - a} + \frac{3 a}{a + 1} - \frac{5}{{a}^{2} - 1}} = \textcolor{b l u e}{\frac{2 {a}^{2} - 4 a - 5}{{a}^{2} - 1}}$

$\textcolor{w h i t e}{\frac{a}{1 - a} + \frac{3 a}{a + 1} - \frac{5}{{a}^{2} - 1}} = \frac{2 {a}^{2} - 2 - 4 a - 3}{{a}^{2} - 1}$

$\textcolor{w h i t e}{\frac{a}{1 - a} + \frac{3 a}{a + 1} - \frac{5}{{a}^{2} - 1}} = \frac{2 \left({a}^{2} - 1\right) - \left(4 a + 3\right)}{{a}^{2} - 1}$

$\textcolor{w h i t e}{\frac{a}{1 - a} + \frac{3 a}{a + 1} - \frac{5}{{a}^{2} - 1}} = \textcolor{b l u e}{2 - \frac{4 a + 3}{{a}^{2} - 1}}$

Sep 4, 2016

$\frac{a}{1 - a} + \frac{3 a}{a + 1} - \frac{5}{{a}^{2} - 1} = 2 + \frac{4 a + 3}{1 - {a}^{2}}$

#### Explanation:

Make the denominator the same for all 3 fractions. After which add up the numerators of the 3 fractions.

$\frac{a}{1 - a} + \frac{3 a}{a + 1} - \frac{5}{{a}^{2} - 1}$

$= \frac{a}{1 - a} \textcolor{b l u e}{\cdot \frac{a + 1}{a + 1}} + \frac{3 a}{a + 1} \textcolor{b l u e}{\cdot \frac{1 - a}{1 - a}} + \frac{5}{1 - {a}^{2}}$

$= \frac{{a}^{2} + a}{1 - {a}^{2}} + \frac{3 a - 3 {a}^{2}}{1 - {a}^{2}} + \frac{5}{1 - {a}^{2}}$

$= \frac{\left({a}^{2} + a\right) + \left(3 a - 3 {a}^{2}\right) + 5}{1 - {a}^{2}}$

$= \frac{- 2 {a}^{2} + 4 a + 5}{1 - {a}^{2}}$

To simplify even more, perform long division.

$\frac{- 2 {a}^{2} + 4 a + 5}{1 - {a}^{2}} = 2 + \frac{4 a + 3}{1 - {a}^{2}}$