The synthetic division and the testing of polynomials

We use the synthetic division when we are solving the polynomial of the form x+c. We need to understand that the “c” is constant, we use the synthetic decision to make the long division easy for us. The synthetic division calculator makes the whole process a little easy for us. We simply need to enter the figure in the calculator to find the synthetic division. For finding the synthetic division the divisor should be a linear shape of x+c. We actually remodel the long division of the polynomial to find the synthetic division. The synthetic division is only applicable when the divisor is of a shape  (x+c) and a first-degree binomial and a leading coefficient of degree “1”. We use the synthetic division to find if the possible is existing for a figure or not.

Why do we use the synthetic division?

When you are finding difficulty dealing with the polynomials, then the synthetic division calculator is the best for the users and for the students. It can be a great source for the students especially to find the perfect roots of long polynomials. You can say goodbye to the stress of solving the long division of the polynomial as the synthesis division is the easiest way to solve the polynomial.

The steps in calculating the synthetic division:

There are general steps to follow when you are going to use the synthetic division methodology on any polynomials:

  • The synthetic division calculator with steps is a great source to turn a polynomial perfect for the synthetic division. The first step is to make sure the polynomial should be written in descending order and spot any missing degree of the x. Make sure the highest degrees come in the first place.
  • A synthetic substitution calculator is great to solve the synthetic division of the polynomials, but you have to write down all the coefficients and the missing term of any degree and the constant values.
  • Place all the roots you are testing outside the synthetic division and the synthetic division sign which should be on the left and bottom side of the rectangle. This representation is specific to the synthetic division. The synthetic substitution calculator can be great to solve the synthetic division for the polynomial
  • You need to drop down the first coefficient below the division sign in the synthetic division.als.
  • You need to multiply the roots, you are going to test for the synthetic division and then drop down the answer below the next coefficient of the synthetic division.
  • Add the coefficient for the synthetic division and also put the answer in the next line. You can use the Polynomial Long Division Calculator for solving the synthetic division of the polynomials.
  • Now you need to multiply the roots you are going to test and need to put the product below the next coefficient.
  • Continue the synthetic division, and multiply and add the coefficient until the last number in the synthetic division sign has been solved.
  • If you are going to get the remainder, then the polynomial you are testing is not a root. If the answer is “0”, then you have just solved the roots and these are the roots you are looking for by the Synthetic division. 
  • The number below the synthetic division signs is the coefficient of the quotient polynomial in the synthetic division. The long division calculator can be used to find the roots of the polynomials by the synthetic division method.

Example of the synthetic division:

Now we try to understand the synthetic division method by the following example:

Consider, you are testing the roots of the equation, f(x)= 2×4- 9×3 – 20×2+ 88x+ 48, and you are going to check the roots x=4 by using the long division method. Now you can also use the Synthetic division calculator, to solve the polynomial for the synthetic division.

  1. Now 4 is outside the figure, and you are testing for the roots. The number inside is the coefficients of the polynomials, here we are going to solve the synthetic division step by step:
  2. We would take the 2 and bring it down and multiply with the roots 4, and then write down the answer, which is, in this case, is “8”, and write down it under the figure “-9”, and when we are adding them the answer we are getting is the “-1”.
  3. Now multiply the “-1”, with the 4, we would get the answer, which is “-4”, and we would write down under the “-21”.
  4. Now we actually add, “-21” and “-4”, we would get the answer, which is “-25”, we would write down under.
  5. Now we would multiply the 4 by “-25”, we would get the answer, which is “-100”.
  6. Now we wrote down the “-100”, under the “88”.
  7. In the next step, we would add “-100” and “88”. The synthetic division can become easy by using the  synthetic division calculator.
  1. The result would be “-12”, which should be written under for the given example.
  2. Now multiply the “-12” by “4”, the result we would get would be “-48” for our synthetic division.
  3. We write down it under the “48” and add it with “-48”, the result would be “0”, in this case, which means the “4” is root.


We have tested the roots of the equation, f(x)= 2×4- 9×3 – 20×2+ 88x+ 48, by the roots x=4, as the result of the synthetic division is equal to the “0”. It means the “4” is the root of the polynomial f(x)= 2×4- 9×3 – 20×2+ 88x+ 48. We can find the synthetic division, with the help of the synthetic division calculator, as it is more than useful, as we can determine the roots of the polynomial and can use them in future calculations. The synthetic division is a simple way to avoid the long division of the polynomials. This can be great for the professional to save their time and space.

Manjusha Nambiar

Hi, I'm Manjusha. This is my blog where I give IELTS preparation tips.