Synthetic Division Calculator

Enter your numerator coefficients below in the corresponding boxes or
x6 x5 x4 x3 x2 x Constant

<-- Select x + or x - and Enter Root (r) <-- This is your denominator

Using synthetic division (Ruffini's Rule), perform the following division:

3x4 + 6x3 - 123x2 - 126x + 1080
x - 5

Determine our root divisor:
To find our root, we solve the divisor equation x - 5 = 0
We add 5 to each side of the equation to get x - 5 + 5 = 0 + 5
Therefore, our root becomes x = 5

Step 1: Write down our coefficients horizontally and our root of 5 to the left:

3 6 -123 -126 1080
5


Step 2: Bring down the first coefficient of 3

3 6 -123 -126 1080
5
3


Step 3: Multiply our root of 5 by our last result of 3 to get 15 and put that in column 2:

3 6 -123 -126 1080
5 15
3


Step 4: Add the new entry of 15 to our coefficient of 6 to get 21 and put this in the answer column 2:

3 6 -123 -126 1080
5 15
3 21


Step 5: Multiply our root of 5 by our last result of 21 to get 105 and put that in column 3:

3 6 -123 -126 1080
5 15 105
3 21


Step 6: Add the new entry of 105 to our coefficient of -123 to get -18 and put this in the answer column 3:

3 6 -123 -126 1080
5 15 105
3 21 -18


Step 7: Multiply our root of 5 by our last result of -18 to get -90 and put that in column 4:

3 6 -123 -126 1080
5 15 105 -90
3 21 -18


Step 8: Add the new entry of -90 to our coefficient of -126 to get -216 and put this in the answer column 4:

3 6 -123 -126 1080
5 15 105 -90
3 21 -18 -216


Step 9: Multiply our root of 5 by our last result of -216 to get -1080 and put that in column 5:

3 6 -123 -126 1080
5 15 105 -90 -1080
3 21 -18 -216


Step 10: Add the new entry of -1080 to our coefficient of 1080 to get 0 and put this in the answer column 5:

3 6 -123 -126 1080
5 15 105 -90 -1080
3 21 -18 -216 0


Our synthetic division is complete. The values in our results row form a new equation, which has a degree 1 less than our original equation shown below:
Leading Answer Term = x(4 - 1) = x3

Since the last number in our result line = 0, we will not have a remainder and have a clean quotient which is shown below in our answer:


Answer = 3x3 + 21x2 - 18x - 216

It appears your answer forms a cubic equation since the maximum power of your result equation is 3 and your remainder is zero. Click here to solve this cubic equation