Polynomial Synthetic Division calculator with steps

Synthetic Division calculator


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If you want to solve a series of synthetic division exercises and need a little help, you have come to the right place. The Synthetic Division Calculator is an excellent tool to study and understand this polynomial division technique based on Ruffini’s rule.

To use the polynomial synthetic division calculator you just have to follow these steps:

  1. First, if the polynomials you are going to divide use a variable other than x, then you must indicate this in the “choose variable” field. You must take into account that you can only perform divisions with polynomials formed of a single variable.
  2. Enter in the first field the polynomial of the dividend in standard form, that is, descending according to the exponents. Ex: x5+3x4+2x2-9x+1
  3. Enter the divisor polynomial in the second field in descending order. Ex: x+3
    Then you will only have to press the «Solve» button and a window will automatically be displayed with the solution and the procedure according to the Ruffini’s rule.

Here is an explanation of the synthetic division method or Ruffini’s rule.

What is synthetic division?

Synthetic division is a simplified method of dividing polynomials. This method is also called division by Ruffini’s rule, since today we can make use of this magnificent method thanks to the study carried out by Paolo Ruffini in the 19th century for divisions of polynomials in which the divisor is of the form x±k.

The synthetic division method ignores the variables, thus focusing on the coefficients. It is important to note that it is possible to use the ruffini rule for divisions in which the divisor is of a higher degree than unity (the synthetic division calculator can do it), but to be more practical in the rest of this article we will focus on the case of a linear divider.

How to do synthetic division of polynomials? | synthetic division steps

To perform synthetic division, you need to have a polynomial and a linear expression in the form of (x – a). The polynomial should be written in descending order of the exponents, and the linear expression should be written as the divisor.

To divide the polynomial by the linear expression, you follow these steps:

  1. Write the coefficients of the polynomial in a row, starting with the coefficient of the term with the highest exponent.

  2. Write the linear expression as the divisor, with a blank space below it.

  3. In the first blank space below the divisor, write the coefficient of the term with the highest exponent in the polynomial.

  4. Multiply the coefficient in the divisor by the coefficient in the first blank space and write the result in the second blank space.

  5. Add the result in the second blank space to the coefficient in the second blank space in the polynomial. Write the result in the third blank space.

  6. Repeat this process until you have filled all the blank spaces.

We will clearly explain how to divide polynomials by the synthetic division method by presenting the following example:

Synthetic division example:
(x3+2x2+7x+6)/(x+3)

  • To solve this operation by synthetic division, we must first set the denominator equal to zero to find the number that we will place in the left part of the division box. In this example, clearing x+3=0, we are left with x=-3, so we must place the value -3 in the left part of the box.
  • Then we must order the dividend polynomial in descending order according to its exponents. Once ordered, we extract only the coefficients and place them in the upper part of the box, remaining as shown in the following image:
Synthetic Division example - step 01
  • Once we have placed all the coefficients in the division box, we lower the main coefficient to the bottom of the box, as shown in the following image:
Synthetic Division example - step 02
  • Then, we multiply the coefficient that we just lowered by the number located to the left of the box, we place the result in the next column just above the lower horizontal line, as shown in the following image:
  • Then we add the numbers that are in the same column and the result is placed in the same column but below the lower horizontal line:
  • The next step will be to multiply the number resulting from the previous step by the number located on the left side of the box, the result will be placed in the next column just above the lower horizontal line.
Synthetic Division example - step 05
  • We add the values that are in the same column and the result is placed in the same column but below the lower horizontal line:
Synthetic Division example - step 06
  • Continuing to apply the sequence of steps given so far, we will have the following final result:
  • Where x2-x+10 is the quotient and -24 is the remainder.
  • The final result is expressed using the formula Quotient + Remainder/Divisor. In our case the solution would be:

x2-x+10 – 24/(x+3)

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