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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.
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:
$$ x^4 + 10x + 1 = \color{blue}{x^3 - 3x^2 + 9x - 17} + \dfrac{ \color{orangered}{52} }{ x + 3 } $$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.