Question

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

Answer 1

`(3r^3+34r^2+89r+75)=(r+8)(3r^2+10r+9)+3`

Explanation:

Here,

`p(r)=3r^3+34r^2+89r+75and` divisor `r=-8`

We take coefficients of `p(r)` and set the problem as shown below.

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

`(-8) |` `color(red)3color(white)(.....)34color(white)(.....)89color(white)(....)75`
`ulcolor(white)((....2))|` `ul(color(red)0color(white)(..................................)`
`color(white)(..........)color(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`

`(-8) |` `3color(white)(.....)color(blue)(34)color(white)(.....)89color(white)(....)75`
`ulcolor(white)((....2))|` `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`

`(-8) |` `3color(white)(.......)34color(white)(.....)color(brown)(89)color(white)(....)75`
`ulcolor(white)((....2))|` `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`

`(-8) |` `3color(white)(.....)34color(white)(.......)89color(white)(........)color(violet)(75`
`ulcolor(white)((....2))|` `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(r)=3r^2+10r+9 and"the Remainder"=3`

Hence ,

`(3r^3+34r^2+89r+75)=(r+8)(3r^2+10r+9)+3`