# How do you divide (x^3+9x^2+23x+15)div(x+5) using synthetic division?

Sep 1, 2016

(x^3+9x^2+23x+15)div(x+5)=color(green)(x^2+4x+3)color(white)("XXX")"Remainder: "color(green)(0)
(see synthetic division method below)

#### Explanation:

Set up a table with the coefficients of the dividend on the first row ($\textcolor{red}{\text{row A}}$ below) leaving a little extra space at the beginning of the line.

Leave a blank line (to be filled in as we go along).

Write the negative of the divisor at the far left of the third line ($\textcolor{red}{\text{row C}}$ below).

You should have something that looks like:
$\textcolor{w h i t e}{\text{XXXX")color(cyan)(x^3)color(white)("XX")color(cyan)(x^2)color(white)("XX")color(cyan)(x^1)color(white)("XX}} \textcolor{c y a n}{{x}^{0}}$
$\textcolor{w h i t e}{\text{XXx")|color(white)("XX")1color(white)("XX")9color(white)("XX")23color(white)("XX")15color(white)("X")color(red)(" row A}}$
underline(color(white)("xxx")|color(white)("XXXXXXXXXXXX"))color(red)(" row B")
$- 5 \textcolor{w h i t e}{\text{x")|color(white)("XXXXXXXXXXXXXX")color(red)("row C}}$

For each coefficient column in turn from left to right:
Add the value in $\textcolor{red}{\text{row B}}$ to the (coefficient) value in $\textcolor{red}{\text{row A}}$ and write the result in $\textcolor{red}{\text{row C}}$
Multiply the negative divisor (on the far left of $\textcolor{red}{\text{row C}}$) by the value just written in $\textcolor{red}{\text{row C}}$ and writer the product in the next column of $\textcolor{red}{\text{row B}}$

After processing the first column, your result should look like:
$\textcolor{w h i t e}{\text{XXx")|color(white)("XX")1color(white)("XX")9color(white)("XX")23color(white)("XX")15color(white)("X")color(red)(" row A}}$
underline(color(white)("XXx")|color(white)("XXxx")-5color(white)("XXXXXXX"))color(red)(" row B")
$- 5 \textcolor{w h i t e}{\text{x")|color(white)("X")1color(white)("XXXXXXXXXXXx")color(red)("row C}}$

...and after processing all the columns, like:
$\textcolor{w h i t e}{\text{XXx")|color(white)("XX")1color(white)("XX")9color(white)("XX")23color(white)("XX")15color(white)("X")color(red)(" row A}}$
underline(color(white)("XXx")|color(white)("XXxx")-5color(white)("x")-20color(white)("x")-15)color(red)(" row B")
$- 5 \textcolor{w h i t e}{\text{x")|color(white)("X")1color(white)("XX")4color(white)("XXX")3color(white)("XX")0color(white)("X")color(red)("row C}}$

The final result in $\textcolor{red}{\text{row C}}$ gives the coefficients of the resulting quotient plus (as the last value in $\textcolor{red}{\text{row C}}$ the remainder:
$- 5 \textcolor{w h i t e}{\text{x")|color(white)("X")color(cyan)(underline(1color(white)("XX")4color(white)("XXX")3))color(white)("XX")color(brown)(0)color(white)("XXXXXXX")color(red)("row C}}$
$\textcolor{w h i t e}{\text{XXXX")color(cyan)(x^2color(white)("XX")x^1color(white)("XX")x^0)color(white)("XX")color(brown)("Remainder}}$