Math Calculators, Lessons and Formulas

It is time to solve your math problem

mathportal.org

Synthetic division calculator

google play badge app store badge

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.

Synthetic Division Calculator
Shows steps on how to divide polynomials.
help ↓↓ examples ↓↓ tutorial ↓↓
(3x^2+3x-5)÷(x+2)
(5x^3-3x+8)÷(x-4)
Divide dividend by the divisor using synthetic division
Find the remainder when dividend is divided by divisor
Determine whether divisor is a factor of dividend
thumb_up 444 thumb_down

Get Widget Code

working...
Examples
ex 1:
Divide 3x3-5x+2 by x-4 using synthetic division.
ex 2:
Find the remainder when 5x^4-2x^3-4x^2+2 is divided by x-2.
ex 3:
Divide -x^5-5x^3-x^2+2 by 3x-1.
ex 4:
Determine whether x-1 is a factor of 3x^3-5x^2-x+3.
Find more worked-out examples in our database of solved problems..
Search our database with more than 300 calculators
TUTORIAL

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.

Example 1 : Divide x2+3x-2 by x-2.

Step 1: Write down the coefficients of 2x2+3x+4 into the division table.

$$ \begin{array}{c|rrr} \color{blue}{\square} &2&3&4\\ & & & \\ \hline & & & \end{array} $$

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.

$$ \begin{array}{c|rrr} \color{blue}{2} &2&3&4\\ & & & \\ \hline & & & \end{array} $$

Step 3: Carry down the leading coefficient

$$ \begin{array}{c|rrr} 2 &\color{orangered}{2}&3&4\\ & & & \\ \hline &\color{orangered}{2}& & \end{array} $$

Step 4: Multiply carry-down by left term and put the result into the next column

$$ \begin{array}{c|rrr} \color{blue}{2} &2&3&4\\ & &\color{blue}{4} & \\ \hline &\color{blue}{2}& & \end{array} $$

Step 5: Add the last column

$$ \begin{array}{c|rrr} 2 &2&\color{orangered}{3}&4\\ & &\color{orangered}{4}& \\ \hline &2&\color{orangered}{7}& \end{array} $$

Step 6: Multiply previous value by left term and put the result into the next column

$$ \begin{array}{c|rrr} \color{blue}{2} &2&3&4\\ & &4&\color{blue}{14} \\ \hline &2&\color{blue}{7}& \end{array} $$

Step 7: Add the last column

$$ \begin{array}{c|rrr} \color{blue}{2} &2&3&\color{orangered}{4}\\ & &4&\color{orangered}{14} \\ \hline &2&7& \color{orangered}{18} \end{array} $$

Step 8: Read the result from the synthetic table.

$$ \begin{array}{c|rrr} 2&2&3&4\\ & &4&14\\ \hline &\color{blue}{2}&\color{blue}{7}& \color{orangered}{18} \end{array} $$

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)

$$ \begin{array}{c|rrr} \color{blue}{\square} &1&0&0& 10&1\\ & & & & &\\ \hline & & & & & \end{array} $$

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.

$$ \begin{array}{c|rrr} \color{blue}{-3}&1&0&0&10&1\\ & & & & &\\ \hline & & & & & \end{array} $$

Step 3: Carry down the leading coefficient

$$ \begin{array}{c|rrr} \color{blue}{-3}&\color{orangered}{1}&0&0&10&1\\ & & & & &\\ \hline &\color{orangered}{1}& & & & \end{array} $$

Multiply carry-down by left term and put the result into the next column

$$ \begin{array}{c|rrr} \color{blue}{-3}&1&0&0&10&1\\ & &\color{blue}{-3}& & &\\ \hline &\color{blue}{1}& & & & \end{array} $$

ADD the last column

$$ \begin{array}{c|rrr} -3 &1&\color{orangered}{0}&0&10&1\\ & &\color{orangered}{-3}& & &\\ \hline &1&-3 & & & \end{array} $$

Multiply last value by left term and put the result into the next column

$$ \begin{array}{c|rrr} \color{blue}{-3} &1&0&0&10&1\\ & &-3&\color{blue}{9}& &\\ \hline &1&\color{blue}{-3} & & & \end{array} $$

ADD the last column

$$ \begin{array}{c|rrr} -3 &1& 0&\color{orangered}{0}&10&1\\ & &-3&\color{orangered}{9}& &\\ \hline &1&-3&\color{orangered}{9}& & \end{array} $$

Multiply last value by left term and put the result into the next column

$$ \begin{array}{c|rrr} \color{blue}{-3} &1& 0&0&10&1\\ & &-3&9& \color{blue}{-27}&\\ \hline &1&-3&\color{blue}{9}& & \end{array} $$

ADD the last column

$$ \begin{array}{c|rrr} -3 &1&0&0&10&\color{orangered}{1}\\ & &-3& 9 & \color{orangered}{-27}&\\ \hline &1&-3&9& \color{orangered}{-17}& \end{array} $$

Multiply last value by left term and put the result into the next column

$$ \begin{array}{c|rrr} \color{blue}{-3} &1&0&0&10&1\\ & &-3& 9 &-27&\color{blue}{51}\\ \hline &1&-3&9&\color{blue}{-17}& \end{array} $$

ADD the last column

$$ \begin{array}{c|rrr} -3 &1&0&0&10&\color{orangered}{1}\\ & &-3& 9 &-27&\color{orangered}{51}\\ \hline &1&-3&9&-17&\color{orangered}{52} \end{array} $$

Step 7: Read the result from the synthetic table.

$$ \begin{array}{c|rrr} -3 &1&0&0&10&\color{orangered}{1}\\ & &-3& 9 &-27&\color{orangered}{51}\\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{9}&\color{blue}{-17}&\color{orangered}{52} \end{array} $$

The quotient is x^3-3x^2+9x-17 and the remainder is 52.

Starting polynomial x^4 + 10x + 1 can be written as:

$$ x^4 + 10x + 1 = \color{blue}{x^3 - 3x^2 + 9x - 17} + \dfrac{ \color{orangered}{52} }{ x + 3 } $$
RESOURCES
443 636 521 solved problems
×
ans:
syntax error
C
DEL
ANS
±
(
)
÷
×
7
8
9
4
5
6
+
1
2
3
=
0
.