# How do you divide (3r^3+34r^2+89r+75)div(r+8) using synthetic division?

##### 1 Answer
Jul 18, 2018

$\left(3 {r}^{3} + 34 {r}^{2} + 89 r + 75\right) = \left(r + 8\right) \left(3 {r}^{2} + 10 r + 9\right) + 3$

#### Explanation:

Here,

$p \left(r\right) = 3 {r}^{3} + 34 {r}^{2} + 89 r + 75 \mathmr{and}$ divisor $r = - 8$

We take coefficients of $p \left(r\right)$ and set the problem as shown below.

color(red)("Put zero {0} below the first number {3} and add : " 3+0=3

$\left(- 8\right) |$ $\textcolor{red}{3} \textcolor{w h i t e}{\ldots . .} 34 \textcolor{w h i t e}{\ldots . .} 89 \textcolor{w h i t e}{\ldots .} 75$
$\underline{\textcolor{w h i t e}{\left(\ldots .2\right)}} |$ ul(color(red)0color(white)(..................................)
$\textcolor{w h i t e}{\ldots \ldots \ldots .} \textcolor{red}{3}$

color(blue)("Now multiply this {3}with divisor" color(blue)((-8)to3xx(-8)=-24 color(blue)("and put below second number{34} and add"

color(blue)(to34+(-24)==10

$\left(- 8\right) |$ $3 \textcolor{w h i t e}{\ldots . .} \textcolor{b l u e}{34} \textcolor{w h i t e}{\ldots . .} 89 \textcolor{w h i t e}{\ldots .} 75$
$\underline{\textcolor{w h i t e}{\left(\ldots .2\right)}} |$ ul(0color(white)(....)color(blue)(-24)color(white)(......................)
color(white)(..........)3color(white)(.....)color(blue)(10color(white)(.....20color(white)(.........)ul|0|
Again repeat the process :

i.e. color(brown)(10xx(-8)=-80 and 89+(-80)=9

$\left(- 8\right) |$ $3 \textcolor{w h i t e}{\ldots \ldots .} 34 \textcolor{w h i t e}{\ldots . .} \textcolor{b r o w n}{89} \textcolor{w h i t e}{\ldots .} 75$
$\underline{\textcolor{w h i t e}{\left(\ldots .2\right)}} |$ ul(0color(white)(...)-24color(white)(..)color(brown)(-80)color(white)(.........10
color(white)(..........)3color(white)(.....)10color(white)(........)color(brown)(9)color(white)(.........ul|0|

Again , color(violet)(9xx(-8)=-72 and (75)+(-72)=3

$\left(- 8\right) |$ 3color(white)(.....)34color(white)(.......)89color(white)(........)color(violet)(75
$\underline{\textcolor{w h i t e}{\left(\ldots .2\right)}} |$ ul(0color(white)(.)-24color(white)(..)-80color(white)(.....)color(violet)(-72
color(white)(..........)3color(white)(......)10color(white)(........)9color(white)(.........)color(violet)(ul|3|
We can see that , quotient polynomial :

$q \left(r\right) = 3 {r}^{2} + 10 r + 9 \mathmr{and} \text{the Remainder} = 3$

Hence ,

$\left(3 {r}^{3} + 34 {r}^{2} + 89 r + 75\right) = \left(r + 8\right) \left(3 {r}^{2} + 10 r + 9\right) + 3$