Synthetic Division – The Complete Technique

Two distinct approaches exist in mathematics for dividing polynomials.

● One is the method of long division.

● The synthetic division approach is another.

Out of these two approaches, the synthetic division method is the quicker way to divide polynomials. When dividing by a linear factor, it is also known as the polynomial division method of a specific situation.  

The long division has been replaced by it. You may find this strategy easier in some circumstances. Even now, you can use the best polynomial synthetic division calculators to solve this particular type of problem in seconds. The tool is loaded with a simple UI that lets you enter a polynomial and a linear factor to get the division answer.

What Is Synthetic Division?

It is a simpler way to divide a polynomial by a linear factor.

With the help of synthetic division, you can instantly find the roots of a division which could be a difficult task if dealing with polynomial long division method.

Mathematical Description:

Suppose there is a polynomial P(x) that you need to divide by a linear factor x-a). A linear factor has a degree of 1. Even the polynomial synthetic division calculator does not give you results if the divisor factor is not linear. So you need to understand its importance and carefully handle the calculations.

Coming to the point again, let us have Q(x) and R as quotient and the remainder. So we have:


P(x)/Q(x) = P(x)/(x-a) = Quotient + [Remainder/(x-a)]

P(x)/(x-a) = Q(x) +[R/(x-a)]

How Can a Synthetic Division Be Performed?

Calculating Synthetic division of polynomials should be done by dividing them by a linear equation, with a 1 serving as the leading coefficient or initial number. Another name for this division by linear denominator computation is division using Ruffini’s rule (calculation with paper and pencil).

The following are the prerequisites needed to use the synthetic process method:

● The polynomial in question should have a degree one divisor. It implies that the specified variable’s exponent ought to be 1. This type of divisor is referred to as a linear factor.

● Additionally, the divisor variable’s coefficient (let’s assume x) needs to equal 1.

Benefits and Demerits of Synthetic Division:

Like every mathematical method, the synthetic division has its advantages and disadvantages including:


● You do not have to perform a long division of polynomials which could be very time-consuming and tricky

● There are no variables that you have to deal with while performing the calculation

● Chances of error are minimal as well


Synthetic division has only one demerit which is as follows:

● The method can be applied to divide polynomials if the divisor is a linear factor, as discussed above

How Does Synthetic Division Calculator Help You?

The tool is developed specifically to solve polynomial division problems by the particular method under study here. The following reasons make the calculator a stand-out tool for solving polynomial division:

● It is super easy to enter the values required

● The tool is easy to use

● It functions very fast

● You can share the results with your education mates

● You can share the link to results in online forums

● You can even save the results in PDF form which is a great plus

Frequently Asked Questions:

What prerequisites does the synthetic division approach have?

The following are prerequisites for the synthetic division method:

● The polynomial expression’s divisor (linear factor) must have a degree of one.

● The variable in the divisor should have a leading coefficient of one.

What Is Synthetic Division’s Primary Use?

The primary application of synthetic division is the determination of polynomial roots’ zeroes.

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