# How do you use synthetic division to divide (14x^2 - 34) / (x + 4)?

Nov 12, 2016

$\setminus \frac{14 {x}^{2} - 34}{x + 4} = 14 x - 56 + \setminus \frac{258}{x - 4}$

## The premise

given factor $\left(x - a\right)$ to $f \left(x\right) = B {x}^{2} + C x + D$,

The pink arrow means multiply $a$ by whatever the difference of the previous column's value was (it's subtraction).
You start out with taking $B$ and putting it in the difference row, so the next column will be $C - a B$. The next column should be $D - a \left(C - a B\right)$, and so on. This applies to all polynomial equations.

Also, you CANNOT miss a term, so if you have something like $3 {x}^{5} + 8 {x}^{3} - 2 {x}^{2}$ you would have to write in 0s for the missing terms...
like this $\setminus \rightarrow 3 {x}^{5} + 0 {x}^{4} + 8 {x}^{3} - 2 {x}^{2} + 0 x - 0$

Your result should have all the powers of $x$ shifted down one degree, so a polynomial beginning with ${x}^{5}$ would have a quotient beginning with ${x}^{4}$.

If the last column does not get you a $0$ as the result, then it will be the remainder, which you put over the factor you divided by, $\setminus \frac{\setminus \textrm{r e m a \in \mathrm{de} r}}{x - a}$

## Actual calculation

$14 {x}^{2} - 34 \setminus \leftrightarrow 14 {x}^{2} + 0 x - 34$
dividing by $\left(x + 4\right)$; from $\left(x - a\right)$ form that means $a = - 4$

So we have the synthetic division set up:
-4 | 14 0 -34

----.--------.------.
$\setminus \textcolor{w h i t e}{a b c}$14

-4 | 14 $\setminus \textcolor{w h i t e}{a}$ 0 $\setminus \textcolor{w h i t e}{a b c}$-34
$-$$\setminus \textcolor{w h i t e}{a b c}$(-56) $\setminus \textcolor{w h i t e}{a}$224
---.----------.----------.
$\setminus \textcolor{w h i t e}{a b c}$14$\setminus \textcolor{w h i t e}{a b}$56$\setminus \textcolor{w h i t e}{a b c}$258

We get $14 x - 56 + \setminus \frac{258}{x - 4}$