Synthetic division calculator
This calculator divides polynomials by binomials using synthetic division. Additionally, the calculator computes the remainder when a polynomial is divided by x−c and checks if the divisor is a factor of dividend. The calculator shows all the steps and provides a full explanation for each step.
Tutorial
What is synthetic division?
Synthetic division is, by far, the easiest and fastest method to divide a polynomial by x-c, where c is a constant. This method only works when we divide by a linear factor. Let's look at two examples to learn how we can apply this method.
Worked examples
Example 1 : Divide x2+3x-2 by x-2.
Step 1: Write the coefficients of 2x2+3x+4 into the division table.
| 2 | 3 | 4 | |
Step 2: Change the sign of a number in the divisor and write it on the left side. In this case, the divisor is x - 2 so we have to change -2 to 2.
| 2 | 2 | 3 | 4 |
Step 3: Carry down the leading coefficient
| 2 | 2 | 3 | 4 |
| 2 |
Step 4: Multiply the previous value by the left term and put the result into the next column.
| 2 | 2 | 3 | 4 |
| 4 | |||
| 2 |
Step 5: Add the last column
| 2 | 2 | 3 | 4 |
| 4 | |||
| 2 | 7 |
Step 6: Multiply previous value by left term and put the result into the next column
| 2 | 2 | 3 | 4 |
| 4 | 14 | ||
| 2 | 7 |
Step 7: Add the last column
| 2 | 2 | 3 | 4 |
| 4 | 14 | ||
| 2 | 7 | 18 |
Step 8: Read the result from the synthetic table.
| 2 | 2 | 3 | 4 |
| 4 | 14 | ||
| 2 | 7 | 18 |
The quotient is 2x + 7 and the remainder is 18.
Starting polynomial x2+3x-2 can be written as:
$$ x^2 +3x - 2 = \color{blue}{2x + 7} + \dfrac{ \color{orangered}{18} }{ x - 2 } $$Example 2 : Divide x4+10x+1 by x + 2.
Step 1: Write down the coefficients of x4-10x+1 into the division table. (Note that this polynomial doesn't have x3 and x2 terms, so these coefficients must be zero)
| 1 | 0 | 0 | 10 | 1 | |
Step 2: Change the sign of a number in the divisor and write it on the left side. In this case, the divisor is x+3 so we have to change +3 to -3.
| -3 | 1 | 0 | 0 | 10 | 1 |
Step 3: Carry down the leading coefficient
| -3 | 1 | 0 | 0 | 10 | 1 |
| 1 |
Multiply carry-down by left term and put the result into the next column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | |||||
| 1 |
ADD the last column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | |||||
| 1 | -3 |
Multiply last value by left term and put the result into the next column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | ||||
| 1 | -3 |
ADD the last column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | ||||
| 1 | -3 | 9 |
Multiply last value by left term and put the result into the next column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | -27 | |||
| 1 | -3 | 9 |
ADD the last column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | -27 | |||
| 1 | -3 | 9 | -17 |
Multiply last value by left term and put the result into the next column.
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | -27 | 51 | ||
| 1 | -3 | 9 | -17 |
ADD the last column
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | -27 | 51 | ||
| 1 | -3 | 9 | -17 | 52 |
Step 7: Read the result from the synthetic table.
| -3 | 1 | 0 | 0 | 10 | 1 |
| -3 | 9 | -27 | 51 | ||
| 1 | -3 | 9 | -17 | 52 |
The quotient is x3-3x2+9x-17 and the remainder is 52.
Starting polynomial x4 + 10x + 1 can be written as:
$$ x^4 + 10x + 1 = \color{blue}{x^3 - 3x^2 + 9x - 17} + \dfrac{ \color{orangered}{52} }{ x + 3 } $$1. Synthetic division — college algebra tutorial.
2. Basic examples on how to apply synthetic division.
3. Video tutorial on how to divide third order polynomial by the monomial.
4. Synthetic division algorithm — step-by-step approach.