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

Dec 22, 2016

$x + 8 \text{ Remainder: } 44$
or (which means the same thing)
$x + 8 + \frac{44}{x - 4}$

#### Explanation:

Using Synthetic division:
$\left(\textcolor{red}{1} {x}^{2} + \textcolor{red}{4} x + \textcolor{red}{12}\right) \div \left(x - \textcolor{b l u e}{4}\right)$

{: (,,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")) :}

Dec 23, 2016

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

#### Explanation:

$\textcolor{w h i t e}{\text{XXXXXXXX}}$Set Up

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

$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$Synthetic Division
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$$\textcolor{w h i t e}{\text{XX")"|"color(white)("X")1color(white)("X")4color(white)("x}} 12$
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$underline("+ |"color(white)("XXXXXXXX"))
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$4color(white)("xx")"|"
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$First row are the coefficients of the dividend.
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$4 in the third row is the constant subtracted
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$from $x$ in the divisor

$\textcolor{w h i t e}{\text{XXXXXXXX}}$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)

$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$Synthetic Division
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$$\textcolor{w h i t e}{\text{XX")"|"color(white)("X")1color(white)("X")4color(white)("x}} 12$
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$underline(+" |"color(white)("XXx")color(white)(4)color(white)("x")color(white)(32))
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$$4 \textcolor{w h i t e}{\text{xx")"|"color(white)("X")1color(white)("X")color(white)(8)color(white)("X}} \textcolor{w h i t e}{44}$
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$Add first dividend coefficient column
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$(1 + "nothing") and write sum in
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$first quotient column

$\textcolor{w h i t e}{\text{XXXXXXXX}}$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"))
$\textcolor{w h i t e}{\text{XXXXXXX}} 8 x + 12$
Multiply most recent divisor term ($1 x$)
by the divisor ($x - 4$) and subtract
the product from the (previous) dividend

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

$\textcolor{w h i t e}{\text{XXXXXXXX}}$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"))
$\textcolor{w h i t e}{\text{XXXXXXX}} 8 x + 12$
$\textcolor{w h i t e}{\text{XXXXXXX}} \underline{8 x - 32}$
$\textcolor{w h i t e}{\text{XXXXXXXXXX}} 44$

$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$Synthetic Division
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$$\textcolor{w h i t e}{\text{XX")"|"color(white)("X")1color(white)("X")4color(white)("x}} 12$
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$underline(+" |"color(white)("XXx")4color(white)("x")32)
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$$4 \textcolor{w h i t e}{\text{xx")"|"color(white)("X")1color(white)("X")8color(white)("X}} 44$
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$Note that the final sum on the 3rd line ($44$)
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$is the Remainder;
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$the other sums are the coefficients of $x$
$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$and the quotient constant, respectively.