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.
Divide using synthetic division $$ ( x^{3}+6 ) \div ( x+2 ) $$
solution
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&1&0&0&6\\& & -2& 4& \color{black}{-8} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{4}&\color{orangered}{-2} \end{array} $$The solution is:
$$ \frac{ x^{3}+6 }{ x+2 } = \color{blue}{x^{2}-2x+4} \color{red}{~-~} \frac{ \color{red}{ 2 } }{ x+2 } $$explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&0&0&6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 1 }&0&0&6\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&0&0&6\\& & \color{blue}{-2} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}-2&1&\color{orangered}{ 0 }&0&6\\& & \color{orangered}{-2} & & \\ \hline &1&\color{orangered}{-2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&0&0&6\\& & -2& \color{blue}{4} & \\ \hline &1&\color{blue}{-2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-2&1&0&\color{orangered}{ 0 }&6\\& & -2& \color{orangered}{4} & \\ \hline &1&-2&\color{orangered}{4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&1&0&0&6\\& & -2& 4& \color{blue}{-8} \\ \hline &1&-2&\color{blue}{4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}-2&1&0&0&\color{orangered}{ 6 }\\& & -2& 4& \color{orangered}{-8} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{4}&\color{orangered}{-2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-2x+4 } $ with a remainder of $ \color{red}{ -2 } $.
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:
Learn how to apply synthetic division in 60 seconds.
Example 3: 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.