# How do you divide (8v^5+43v^4+5v+20)div(v+4) using synthetic division?

Aug 10, 2018

$\frac{8 {v}^{5} + 43 {v}^{4} + 5 v + 20}{v + 4}$=$\left(8 {v}^{4} + 11 {v}^{3} - 44 {v}^{2} + 176 v - 699\right) + \frac{2816}{v + 4}$

#### Explanation:

$\left(8 {v}^{5} + 43 {v}^{4} + 5 v + 20\right) \div \left(v + 4\right)$

We can divide this polynomial by using synthetic division

We have , $p \left(v\right) = \left(8 {v}^{5} + 43 {v}^{4} + 0 {v}^{3} + 0 {v}^{2} + 5 v + 20\right)$

$\mathmr{and} \text{divisor :} v = - 4$

We take ,coefficients of $p \left(v\right) \to 8 , 43 , 0 , 0 , 5 , 20$

$- 4 |$ $8 \textcolor{w h i t e}{\ldots \ldots .} 43 \textcolor{w h i t e}{\ldots \ldots \ldots} 0 \textcolor{w h i t e}{\ldots \ldots \ldots .} 0 \textcolor{w h i t e}{\ldots \ldots \ldots .} 5 \textcolor{w h i t e}{\ldots \ldots \ldots .} 20$
$\underline{\textcolor{w h i t e}{\ldots .}} |$ ul(0color(white)(..)-32color(white)(...)-44color(white)(.....)176color(white)(..)-704color(white)(.....)2796
color(white)(......)8color(white)(.......)11color(white)(....)-44color(white)(.....)176color(white)(..)-699color(white)(...)color(white)(..)color(violet)(ul|2816|
We can see that , quotient polynomial :

$q \left(v\right) = 8 {v}^{4} + 11 {v}^{3} - 44 {v}^{2} + 176 v - 699$

$\mathmr{and} \text{the Remainder} = 2816$

Hence ,

$\frac{8 {v}^{5} + 43 {v}^{4} + 5 v + 20}{v + 4} =$

$\left(8 {v}^{4} + 11 {v}^{3} - 44 {v}^{2} + 176 v - 699\right) + \frac{2816}{v + 4}$