# What is synthetic division?

Jan 14, 2016

Synthetic division is a way to divide a polynomial by a linear expression.

#### Explanation:

Suppose our problem is this: $y = {x}^{3} + 2 {x}^{2} + 3 x - 6$

Now, the main use of synthetic division is to find the roots or solutions to an equation.

The process for this serves to cut down on the gessing you have to do to find a value of x that makes the equation equal 0.

First, list the possible rational roots, by listing the factors of the constant (6) over the list of the factors of the lead coefficient (1).

$\pm$(1,2,3,6)/1

Now, you can begin trying numbers. First, you simplify the equation to just the coefficients:
)¯¯1¯¯¯2¯¯¯¯3¯¯¯¯-6¯¯

And now, plug your possible rational roots in, one at a time, until one works. (I suggest doing 1 and -1 first, since they're the easiest)

1 )¯¯1¯¯¯2¯¯¯¯3¯¯¯¯-6¯¯

color(white)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

1 .First bring down the lead number (1)

1 )¯¯1¯¯¯2¯¯¯¯3¯¯¯¯-6¯¯

color(white)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
$\textcolor{w h i t e}{00}$1

2 . Now multiply that number by the divisor (1)

1 )¯¯1¯¯¯2¯¯¯¯3¯¯¯¯-6¯¯

color(white)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
$\textcolor{w h i t e}{00}$1

3 . Now place the product underneath the second number (2)

1 )¯¯1¯¯¯2¯¯¯¯3¯¯¯¯-6¯¯
$\textcolor{w h i t e}{\ddots}$$\textcolor{w h i t e}{00}$1
color(white)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
$\textcolor{w h i t e}{00}$1$\textcolor{w h i t e}{00}$

4 . Now add the two numbers together (2&1) and move the sum down

1 )¯¯1¯¯¯2¯¯¯¯3¯¯¯¯-6¯¯
$\textcolor{w h i t e}{\ddots}$$\textcolor{w h i t e}{00}$1
color(white)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
$\textcolor{w h i t e}{\sum}$1$\textcolor{w h i t e}{00}$3

5 . Now multiply the sum (3) by the divisor (1) and move it underneath the next value in the dividend

1 )¯¯1¯¯¯2¯¯¯3¯¯-6¯¯
$\textcolor{w h i t e}{\ddots}$$\textcolor{w h i t e}{00}$1$\textcolor{w h i t e}{00}$3
color(white)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
$\textcolor{w h i t e}{\sum}$1$\textcolor{w h i t e}{00}$3

6 . Now add the two values together (3&3) and move the sum down

1 )¯¯1¯¯¯2¯¯¯3¯¯-6¯¯
$\textcolor{w h i t e}{\ddots}$$\textcolor{w h i t e}{00}$1$\textcolor{w h i t e}{00}$3
color(white)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
$\textcolor{w h i t e}{\sum}$1$\textcolor{w h i t e}{00}$3$\textcolor{w h i t e}{00}$6

7 . Now multiply the new sum (6) with the divisor (1) and move it underneath the next value in the dividend

1 )¯¯1¯¯¯2¯¯¯3¯¯-6¯¯
$\textcolor{w h i t e}{\ddots}$$\textcolor{w h i t e}{00}$1$\textcolor{w h i t e}{00}$3$\textcolor{w h i t e}{00}$6
color(white)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
$\textcolor{w h i t e}{\sum}$1$\textcolor{w h i t e}{00}$3$\textcolor{w h i t e}{00}$6

8 . Now add together the two values (6&-6) and move that sum down

1 )¯¯1¯¯¯2¯¯¯3¯¯-6¯¯
$\textcolor{w h i t e}{\ddots}$$\textcolor{w h i t e}{00}$1$\textcolor{w h i t e}{00}$3$\textcolor{w h i t e}{00}$6
color(white)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
$\textcolor{w h i t e}{\sum}$1$\textcolor{w h i t e}{00}$3$\textcolor{w h i t e}{00}$6$\textcolor{w h i t e}{00}$0

8 . Now you have the equation, 0=${x}^{2} + 3 x + 6$, with the sums you found being the coeffiecients

1 )¯¯1¯¯¯2¯¯¯3¯¯-6¯¯
$\textcolor{w h i t e}{\ddots}$$\textcolor{w h i t e}{00}$1$\textcolor{w h i t e}{00}$3$\textcolor{w h i t e}{00}$6
color(white)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
$\textcolor{w h i t e}{\sum}$1$\textcolor{w h i t e}{00}$3$\textcolor{w h i t e}{00}$6$\textcolor{w h i t e}{00}$0