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.
Find the resultant polynomial equation using the synthetic division method for p(x) : 3x4 + 2x3 - 123x2 -126x + 1080 and q(x): x + 2 .
p(x): 3x4 + 2x3 - 123x2 -126x + 1080
q(x): x + 2
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 |
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) |
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 |
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 |
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) |
-2 |
3 |
2 |
-123 |
-126 |
1080 | ||||
0 |
-6 |
8 |
230 |
-208 (-2 * 104) | |||||
3 |
-4 |
-115 |
104 |
872 (1080 -208) |
-2 |
3 |
2 |
-123 |
-126 |
1080 | ||||
0 |
-6 |
8 |
230 |
-208 | |||||
3 |
-4 |
-115 |
104 |
872 |
Therefore, the polynomial value of the given p(x) as calculated using the synthetic division example is 3x3 - 4x2 - 115x + 104 + 872 / (x +2)