Synthetic Division of Polynomials

There are two different methods to divide the polynomials in Mathematics. One is the long division method and the second one is the synthetic division method. Among these two methods, the synthetic division method is the shortcut method to divide polynomials. It is also known as the polynomial division method of a special case when it is divided by the linear factor.  The synthetic division replaces the long division method.  In certain situations, you will find this method easier. In this article, we will discuss what synthetic division method is and how to perform synthetic division method along with steps with 

What is the Synthetic Division of Polynomials?

  • The Synthetic division of polynomials is a shortcut of polynomial division, especially if we need to divide by a linear factor.

  •  Synthetic division method is basically used to find out the zeros or roots of polynomials and this method is not for the division of factors. Thus, the formal definition of synthetic division can be defined as:

“Synthetic division of polynomials can be defined as a simplified way of dividing a polynomial with another polynomial equation of degree 1 and this method is generally used to find the zeros of polynomials”

This division method (synthetic division method of polynomials) is performed manually with less effort of calculation than the long division method which requires a lot of calculation. Usually, a binomial term is used as a divisor in this division method, such as x – c.

Synthetic Division Steps (synthetic division method of polynomials)-

STEP 1

  • To begin the problem, set the denominator = zero, to find the number we need to put in the division box.

  • Then, we need to write the numerator is written in descending order and if any of the terms are missing then we need to use a zero to fill in the term which is missing.

  • Finally, we need to list only the coefficient in the division problem.

STEP 2

  • In step 2, when the problem is set up, all we do is bring the first number or the leading coefficient straight down.

STEP 3

  • In step 3, we put the result in the next column by multiplying the number in the division box with the number we brought down in the previous step.

STEP 4

  • In step 4, write the result in the bottom of the row by adding the two numbers we have together.

STEP 5

  • Until we reach the end of the problem, repeat step 3 and step 4.

STEP 6

  • Now we have the answer, write the final answer.

  • The numbers in the bottom row with the last number being the remainder and the remainder which is written as a fraction gives us the final answer.

  • The variables should start with one power less than the real denominator and should go down one with each term.


Here are synthetic method examples for better understanding!

Divide x2+5x +6 by x-1 using the concept of synthetic division.

Let’s go step by step,

Synthetic Division Steps-

First, write the coefficients inside an upside-down division symbol. Here the coefficients are 1 as in x2, 5 as in x and 6.

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Put the test zero, here x = 1, at the left as shown in the picture.

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Now you need to take the first number inside, which represents  the leading coefficient, and now carry it down, unchanged, to below the division symbol in the same way as shown in the picture:

  

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Multiply this carry-down value by the test zero, and carry the result up into the next column as shown  in the picture given below :

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Add down the column just as shown here:

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Multiply the previous carry-down value by the test zero, and carry the new result up into the last column as shown in the picture given on the right:

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 Now add down the column:

This last carry-down value will give you the remainder.

     

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Now, solve the same question. Comparing, here through the synthetic division of polynomial steps can see that you got the same result from the synthetic division, the same quotient (that is, 1x + 6) and the same remainder at the end (which is, 12). You will get the same answer if you perform long division for the same question.

The results we have got are formatted differently, but we should recognize that each format provided us with the result, being a quotient of x + 6, we always get a remainder of 12.

Why the Synthetic Division Method?

Synthetic division is generally known as a shorthand, or shortcut, a method of polynomial division in the special case of dividing by a linear factor and synthetic division works only in this case. Synthetic division method is generally used, however, not for dividing out factors but for finding zeros or roots of polynomials. The method is easier than the long division method for solving synthetic division problems and that’s why it is preferred. 

FAQ (Frequently Asked Questions)

Q1. How to do the Synthetic Division?

Ans. Synthetic division can be defined as another way to divide a polynomial by the binomial x - c, where c is a constant. Let’s know how to do synthetic division.

Synthetic Division of Polynomial Steps-

Step 1) Set up the synthetic division. 

Step 2) Next, you need to bring down the leading coefficient to the bottom row.

Step 3) Now multiply c by the value just written on the bottom row. 

Step 4) Add the column created in step 3.

These are the few steps that are necessary to solve synthetic division problems.


Q2. Who Invented Synthetic Division and what is a Quotient in Synthetic Division Problems?

Ans. Synthetic division method was invented by Paolo Ruffini who took part in a competition that was organized by the Italian Scientific Society (of forty). The question to be answered was a method to find the roots of any polynomial and he got this method.

In synthetic division, P(x) is equal to the dividend, Q(x) is equal to the quotient, and R(x) is equal to the remainder. Let’s see how to do this problem by synthetic division. First, to use synthetic division, the divisor must be of the first degree and it must be in the form x − a.