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

Solution:
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)

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