# How do you divide (2v^3-20v^2+56v-46)div(v-6) using synthetic division?

Jun 11, 2017

The remainder is $= 2$ and the quotient is $= 2 {v}^{2} - 8 v + 8$

#### Explanation:

Let's perform the synthetic division

$\textcolor{w h i t e}{a a a a}$$6$$\textcolor{w h i t e}{a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$2$$\textcolor{w h i t e}{a a a a a a}$$- 20$$\textcolor{w h i t e}{a a a a a a}$$56$$\textcolor{w h i t e}{a a a a a}$$- 46$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$_________

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a a}$$12$$\textcolor{w h i t e}{a a a a}$$- 48$$\textcolor{w h i t e}{a a a a a a a}$$48$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$________

$\textcolor{w h i t e}{a a a a}$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a}$$2$$\textcolor{w h i t e}{a a a a a a}$$- 8$$\textcolor{w h i t e}{a a a a a a a}$$8$$\textcolor{w h i t e}{a a a a a a a a}$$\textcolor{red}{2}$

The remainder is $= 2$ and the quotient is $= 2 {v}^{2} - 8 v + 8$

$\frac{2 {v}^{3} - 20 {v}^{2} + 56 v - 46}{v - 6} = 2 {v}^{2} - 8 v + 8 + \frac{2}{v - 6}$

Jun 11, 2017

$2 {v}^{2} - 8 v + 8 \text{ remainder } 2$

Or $\text{ } 2 {v}^{2} - 8 v + 8 + \frac{2}{v - 6}$

#### Explanation:

In synthetic division, you use only the coefficients, making sure that all powers of $v$ are considered:

$\textcolor{red}{2} {v}^{3} \textcolor{red}{- 20} {v}^{2} \textcolor{red}{+ 56} {v}^{1} \textcolor{red}{- 46} {v}^{0}$

Make $v - 6 = 0 \text{ "rarr v = color(blue)(6)" } \leftarrow$ this is written 'outside'.

$\textcolor{w h i t e}{\times x . x} | \textcolor{red}{2} \text{ "color(red)(-20)" " color(red)(+56)" } \textcolor{red}{- 46}$
$\textcolor{w h i t e}{\times . \times} | \downarrow$
color(white)(xx.x)color(blue)(6)|ul(color(white)(xxxxx)12" "-48" "48
$\textcolor{w h i t e}{\times \times \times} \textcolor{red}{2} \text{ "-8" "+8" } 2 \leftarrow$ the remainder

Steps are as follows:

Bring down the $\textcolor{red}{2}$

Multiply $\textcolor{b l u e}{6} \times \textcolor{red}{2} = 12$

Add $- 20 + 12 = - 8$

Multiply $\textcolor{b l u e}{6} \times - 8 = - 48$

Add $+ 56 - 48 = + 8$

Multiply $\textcolor{b l u e}{6} \times 8 = 48$

Add $- 46 + 48 = + 2$

What do the numbers tell us?

$2 {v}^{3} \div v = 2 {v}^{2} \text{ } \leftarrow$ this is the first term in the quotient.

The numbers represent the co-efficients of the following terms:

$\text{ "color(red)(2)" "-8" "+8" } 2 \leftarrow$ the remainder

$\text{ "2v^2" "-8v" "+8" remainder} 2$

This is the quotient.