Synthetic Division Calculator (2024)

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Synthetic Division Calculator (1)Synthetic Division Calculator (2)

This calculator divides polynomials by binomials using synthetic division.Additionally, the calculator computes the remainder when a polynomial is divided by x−c and checksif the divisor is a factor of dividend.The calculator shows all the steps and provides a full explanation for each step.


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example 1:ex 1:

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

example 2:ex 2:

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

example 3:ex 3:

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

example 4:ex 4:

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

Find more worked-out examples in the database of solved problems..

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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:

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


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.

439 588 094 solved problems

Synthetic Division Calculator (2024)


What is 24x3 − 16x2 20x divided by 4x? ›

Final answer:

To divide the polynomial 24x³ - 16x² + 20x by 4x, use polynomial long division to get the quotient 6x² - 4x + 5.

Is synthetic division easier? ›

The advantages of synthetic division are that it allows one to calculate without writing variables, it uses few calculations, and it takes significantly less space on paper than long division.

How do you find zeros in synthetic division? ›

Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate. Repeat step two using the quotient found from synthetic division.

What is the rule for synthetic division? ›

The process starts by bringing down the leading coefficient. We then multiply it by the “divisor” and add, repeating this process column by column until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder.

Do you add or multiply in synthetic division? ›

We can perform the synthetic division method, only if the divisor is a linear factor. In the synthetic division method, we will perform multiplication and addition, in the place of division and subtraction, which is used in the long division method.

When can you not use synthetic division? ›

Synthetic division can only be used if you're dividing by a LINEAR factor in the form x - a. For example, it works if you are dividing by x - 3 or by x + 4. If you need to divide by a polynomial with a higher degree or one that doesn't have a 1 as the leading coefficient, you'll have to do plain old long division.

What is another name for synthetic division? ›

synthetic division, short method of dividing a polynomial of degree n of the form a0xn + a1xn 1 + a2xn 2 + … + an, in which a0 ≠ 0, by another of the same form but of lesser degree (usually of the form x − a). Based on the remainder theorem, it is sometimes called the method of detached coefficients.

How do you know when to use long or synthetic division? ›

Synthetic division is another method of dividing polynomials. It is a shorthand of long division that only works when you are dividing by a polynomial of degree 1. Usually the divisor is in the form ( x ± a ) . In synthetic division, unlike long division, you are only concerned with the coefficients in the polynomials.

What is the synthetic method in math? ›

The word “synthetic' is derived from the word 'synthesis which means to combine together. In this method we combine together a number of facts, perform certain mathematical operations and arrive at the solution. In this method we start with the known data and connect it with the unknown part.

What is a synthetic equation? ›

: a simplified method for dividing a polynomial by another polynomial of the first degree by writing down only the coefficients of the several powers of the variable and changing the sign of the constant term in the divisor so as to replace the usual subtractions by additions.

How to determine end behavior? ›

To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x gets very large or very small, so its behavior will dominate the graph.

What is the maximum value of this function f x )= − 16x2 32x 20? ›

The final answer is 36 36 . Use the x and y values to find where the maximum occurs.

What is the factor of 4x 20x 25? ›

Factorise the following : 4x² + 20x + 25

We have to factorise the polynomial. Therefore, the factors are (2x + 5) and (2x + 5).

What is the remainder when f x )= 4x 3 20x 50? ›

Summary: The remainder when f(x) = 4x3 - 20x - 50 is divided by (x - 3) is -2.


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