Synthetic Division Calculator (2024)

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  Complex Number Division Calculator + Online Solver With Free StepsHow Do You Find The Average Of Percentages? - ZippiaDividing Monomials Calculator + Online Solver With Free StepsTime Card Calculator

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Divide $3x^3-5x+2$ by $x-4$ using synthetic division.

Find the remainder when $5x^4-2x^3-4x^2 + 2$ is divided by $x-2$.

Divide $-x^5-5x^3-x^2+2$ by $3x-1$.

Determine whether $x-1$ is a factor of $3x^3-5x^2-x+3$.

Synthetic division

Synthetic division is, by far, the easiest and fastest method to divide a polynomial by $ \color{blue}{x - c} $, where $ \color{blue}{c} $ is a constant. This method only workswhen we divide by a linear factor. Let's look at two examples to learn how we can apply this method.

Example 1 : Divide $ x^2 +3x - 2 $ by $x - 2$.

Step 1: Write down the coefficients of $ 2x^2 +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, thedivisor is $x - 2$ so we have to change $\, -2 \,$ to $\, \color{blue}{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 6: 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 7: 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 $ \color{blue}{2x + 7}$ and the remainder is $\color{orangered}{18}$.

Starting polynomial $ x^2 +3x - 2 $ can be written as:

$$ x^2 +3x - 2 = \color{blue}{2x + 7} + \dfrac{ \color{orangered}{18} }{ x - 2 } $$

Example 2 : Divide $ x^4 + 10x + 1 $ by $x + 2$.

Step 1: Write down the coefficients of $ x^4 - 10x + 1 $ into the division table.(Note that this polynomial doesn't have $x^3$ and $x^2$ 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, thedivisor is $x + 3$ so we have to change $\, +3 \,$ to $\, \color{blue}{-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 $ \color{blue}{x^3 - 3x^2 + 9x - 17}$ and the remainder is $\color{orangered}{52}$.

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

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