# Synthetic Division Examples

Find here a few examples to find the polynomial equation using synthetic division method. These synthetic division examples explains how to find the roots of 4th degree polynomials.

## Synthetic Division for 4th Degree Polynomials

###### Example :

Find the resultant polynomial equation using the synthetic division method for p(x) : 3x4 + 2x3 - 123x2 -126x + 1080 and q(x): x + 2 .

###### Given :

p(x): 3x4 + 2x3 - 123x2 -126x + 1080
q(x): x + 2

###### Step 1: Bring the first coefficient down below the horizontal line. (i.e) [ 2 x 0 = 0] Adding 3 and 0, gives 3.
 -2 3 2 -123 -126 1080 0 3
###### Step 2:

Multiply the divisor (here -2) with the number below horizontal line (here 3) and write it down below the next coefficient.
(2 x - 3 ) = -6

 -2 3 2 -123 -126 1080 0 -6 3 -4 (2 - 6)
###### Step 3 :

Add 2 and -6 and write down the next coefficient below the horizontal line.
2 - 6 = -4

 -2 3 2 -123 -126 1080 0 -6 3 -4
###### Step 5:

Multiplying divisor -2 and -4, we get 8.
Write down 8 below the next coefficient.
Again adding 8 and -123, we get - 115.

 -2 3 2 -123 -126 1080 0 -6 8 3 -4 -115
###### Step 6:

Follow the previous steps for all coefficients,

 -2 3 2 -123 -126 1080 0 -6 8 230 (-115 * -2) 3 -4 -115 104 (-126 + 230)
###### Step 7:
 -2 3 2 -123 -126 1080 0 -6 8 230 -208 (-2 * 104) 3 -4 -115 104 872 (1080 -208)
###### Step 8:
 -2 3 2 -123 -126 1080 0 -6 8 230 -208 3 -4 -115 104 872
###### Result:

Therefore, the polynomial value of the given p(x) as calculated using the synthetic division example is 3x3 - 4x2 - 115x + 104 + 872 / (x +2)