# How do you simplify \frac { a ^ { 2} - a - 21} { a - 5}?

##### 1 Answer
Jan 27, 2018

$a + 4 - \frac{1}{a - 5}$

#### Explanation:

The numerator cannot be factored, since there are no two factors of $- 21$ which add up to $- 1$. Therefore, the only way to simplify this is through long division:

$\frac{{a}^{2} - a - 21}{a - 5}$

First, separate the $- a$ into $- 5 a + 4 a$.

$\frac{{a}^{2} - 5 a + 4 a - 21}{a - 5}$

Now, the first two terms can be removed and simplified:

$\frac{{a}^{2} - 5 a}{a - 5} + \frac{4 a - 21}{a - 5}$

$a + \frac{4 a - 21}{a - 5}$

Next, we need to do the same 'splitting' technique with the ones term, in order to be able to factor it out along with the $4 a$:

$a + \frac{4 a - 20 - 1}{a - 5}$

$a + \frac{4 a - 20}{a - 5} - \frac{1}{a - 5}$

$a + 4 - \frac{1}{a - 5}$

There aren't any more terms to "split" and factor out, so this is as simple as we can get.

Final Answer