Question

How do you divide `(x^2+4x+12)div(x-4)`?

Answer 1

`x+8" Remainder: "44`
or (which means the same thing)
`x+8 +(44)/(x-4)`

Explanation:

Using Synthetic division:
`(color(red)1x^2+color(red)4x+color(red)(12)) div (x-color(blue)(4))`

`{: (,,underline(color(red)(""(x^2))),,underline(color(red)(""(x))),,underline(color(red)(" "))), (,,color(red)1,,color(red)4,,color(red)12), (underline(" + ")," | ",underline(color(white)("XX")),,underline(color(blue)4xxcolor(green)1),,underline(color(blue)4xxcolor(orange)8)), (color(blue)(4)," | ",color(green)1,,color(orange)8,,color(magenta)(44)), (,,color(green)(""(x)),,,,color(magenta)("Remainder")) :}`

Answer 2

The following is an attempt to expand on the Synthetic Division process and to compare it to Long Division.

Explanation:

`color(white)("XXXXXXXX")`Set Up

Long Division
`color(white)("XXXx")underline(color(white)("xxxxxxxxxxxxx"))`
`x-4 ) x^2+4x+12`

`color(white)("XXXXXXXXXXXX")`Synthetic Division
`color(white)("XXXXXXXXXXXX")` `color(white)("XX")"|"color(white)("X")1color(white)("X")4color(white)("x")12`
`color(white)("XXXXXXXXXXXX")` `underline("+ |"color(white)("XXXXXXXX"))`
`color(white)("XXXXXXXXXXXX")` `4color(white)("xx")"|"`
`color(white)("XXXXXXXXXXXX")`First row are the coefficients of the dividend.
`color(white)("XXXXXXXXXXXX")`4 in the third row is the constant subtracted
`color(white)("XXXXXXXXXXXX")`from `x` in the divisor

`color(white)("XXXXXXXX")`Division of First Term

Long Division
`color(white)("XXXx")underline(1xcolor(white)("x")color(white)("XXxxxxx"))`
`x-4 ) x^2+4x+12`
Divide the first term of the divisor into the first term
of the dividend and write the result as the first term
of the quotient (above the line)

`color(white)("XXXXXXXXXXXX")`Synthetic Division
`color(white)("XXXXXXXXXXXX")` `color(white)("XX")"|"color(white)("X")1color(white)("X")4color(white)("x")12`
`color(white)("XXXXXXXXXXXX")` `underline(+" |"color(white)("XXx")color(white)(4)color(white)("x")color(white)(32))`
`color(white)("XXXXXXXXXXXX")` `4color(white)("xx")"|"color(white)("X")1color(white)("X")color(white)(8)color(white)("X")color(white)(44)`
`color(white)("XXXXXXXXXXXX")`Add first dividend coefficient column
`color(white)("XXXXXXXXXXXX")`(1 + "nothing") and write sum in
`color(white)("XXXXXXXXXXXX")`first quotient column

`color(white)("XXXXXXXX")`Generate New Dividend Terms

Long Division
`color(white)("XXXx")underline(1xcolor(white)("x")color(white)(+8)color(white)("xxxxx"))`
`x-4 ) x^2+4x+12`
`color(white)("XXXx")underline(x^2color(white)("x")-4xcolor(white)("xxxx"))`
`color(white)("XXXXXXX")8x+12`
Multiply most recent divisor term (`1x`)
by the divisor (`x-4`) and subtract
the product from the (previous) dividend

`color(white)("XXXXXXXXXXXX")`Synthetic Division
`color(white)("XXXXXXXXXXXX")` `color(white)("XX")"|"color(white)("X")1color(white)("X")4color(white)("x")12`
`color(white)("XXXXXXXXXXXX")` `underline(+" |"color(white)("XXx")4color(white)("x"color(white)(32))`
`color(white)("XXXXXXXXXXXX")` `4color(white)("xx")"|"color(white)("X")1color(white)("X")8color(white)("X")color(white)(44)`
`color(white)("XXXXXXXXXXXX")`Multiply the divisor constant (`4`)
`color(white)("XXXXXXXXXXXX")`by the most recent quotient coefficient (`1`)
`color(white)("XXXXXXXXXXXX")`and write the result on row 2
`color(white)("XXXXXXXXXXXX")`under the second dividend coefficient;`color(white)("XXXXXXXXXXXX")`then add the second column to get the next`color(white)("XXXXXXXXXXXX")`quotient term coefficient.

`color(white)("XXXXXXXX")`Repeat Process Until All Terms Used

Long Division
`color(white)("XXXx")underline(1xcolor(white)("x")+8color(white)("xxxxx"))`
`x-4 ) x^2+4x+12`
`color(white)("XXXx")underline(x^2color(white)("x")-4xcolor(white)("xxxx"))`
`color(white)("XXXXXXX")8x+12`
`color(white)("XXXXXXX")underline(8x-32)`
`color(white)("XXXXXXXXXX")44`

`color(white)("XXXXXXXXXXXX")`Synthetic Division
`color(white)("XXXXXXXXXXXX")` `color(white)("XX")"|"color(white)("X")1color(white)("X")4color(white)("x")12`
`color(white)("XXXXXXXXXXXX")` `underline(+" |"color(white)("XXx")4color(white)("x")32)`
`color(white)("XXXXXXXXXXXX")` `4color(white)("xx")"|"color(white)("X")1color(white)("X")8color(white)("X")44`
`color(white)("XXXXXXXXXXXX")`Note that the final sum on the 3rd line (`44`)
`color(white)("XXXXXXXXXXXX")`is the Remainder;
`color(white)("XXXXXXXXXXXX")`the other sums are the coefficients of `x`
`color(white)("XXXXXXXXXXXX")`and the quotient constant, respectively.