# How do you simplify and restricted value of (a^2 - a - 2)/(a^2 - 13a + 30)?

May 19, 2017

$\frac{\left(a + 1\right) \left(a - 2\right)}{\left(a - 3\right) \left(a - 10\right)}$; $a \ne 3 , 10$

#### Explanation:

We have: $\frac{{a}^{2} - a - 2}{{a}^{2} - 13 a + 30}$

Let's factorise both the numerator and the denominator using the "middle-term break":

$= \frac{{a}^{2} + a - 2 a - 2}{{a}^{2} - 3 a - 10 a + 30}$

$= \frac{a \left(a + 1\right) - 2 \left(a + 1\right)}{a \left(a - 3\right) - 10 \left(a - 3\right)}$

$= \frac{\left(a + 1\right) \left(a - 2\right)}{\left(a - 3\right) \left(a - 10\right)}$

Now let's evaluate the restricted values of $a$.

The denominator of the fraction can never be equal to zero:

$R i g h t a r r o w \left(a - 3\right) \left(a - 10\right) \ne 0$

Using the null factor law:

$R i g h t a r r o w a - 3 \ne 0 , a - 10 \ne 0$

$R i g h t a r r o w a \ne 3 , a \ne 10$

$\therefore \frac{{a}^{2} - a - 2}{{a}^{2} - 13 a + 30} = \frac{\left(a + 1\right) \left(a - 2\right)}{\left(a - 3\right) \left(a - 10\right)}$; $a \ne 3 , 10$