Synthetic division

What is Synthetic Division?

The Synthetic division is a shortcut way of polynomial division, especially if we need to divide it by a linear factor. It is generally used to find out the zeroes or roots of polynomials and not for the division of factors. Thus, the definition of synthetic division is:

Synthetic division can be defined as a simplified way of dividing a polynomial with another polynomial equation of degree 1 and is generally used to find the zeroes of polynomials.

This division method is performed manually with less effort of calculation than the long division method. Usually, a binomial term is used as a divisor in this method, such as x – b.

How to do a Synthetic Division?

If we want to divide polynomials using synthetic division, you should be dividing it by a linear expression and the first number or the leading coefficient should be a 1. This division by linear denominator is also called division through Ruffini’s rule(paper-and-pencil computation).

For example, we can use synthetic division method to divide a polynomial of 2 degrees by x + a or x – a, but you cannot use this method to divide by x2 + 3 or 5x2 – x + 7.

If the leading coefficient is not 1, then we need to divide by the leading coefficient to turn the leading coefficient into 1. For example, 4x – 1 would become x – ¼ and 4x+9 would become x + 9/4. If the synthetic division is not working, then we need to use long division.

Synthetic Division Method (Steps)

Following are the steps required for Synthetic Division of a Polynomial:

Step 1
  • To set up the problem, we need to set the denominator = zero, to find the number to put in the division box.
  • Then, the numerator is written in descending order and if any terms are missing we need to use a zero to fill in the missing term.
  • At last, list only the coefficient in the division problem.
Step 2 Now, when the problem is set up perfectly, bring the first number or the leading coefficient straight down.
Step 3 Then, put the result in the next column by multiplying the number in the division box with the number you brought down.
Step 4 Write the result in the bottom of the row by adding the two numbers together and
Step 5 Until you reach the end of the problem, repeat the steps 3 and 4.
Step 6

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 makes the final answer.

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

Synthetic Division Solved Examples and Problems

Example 1:

Divide : \(\frac{2x^{3} – 5x^{2} + 3x + 7}{x-2}\)

Solution:

Following the steps as per explained above to divide the polynomials, we can get;

Synthetic Division Example 1

Synthetic Division Example 1

Example 2: 

Divide : \(\frac{2x^{3} + 5x^{2} + 9}{x+3}\)

Solution:

As per the given question; we have two polynomial in numerator and denominator. Denominator consists of linear equation, so we can easily apply the synthetic division method here.

Follow the step by step method as given below:

Synthetic Division Example 2

Synthetic Division Example 2

Example 3:

Divide : \(\frac{3x^{3} + 5x – 1}{x+1}\)

Solution:

Following the same steps as per previous examples.

Synthetic Division Example 3

Synthetic Division Example 3

Example 4:

Divide : \(\frac{4x^{3} – 8x^{2} -x +5}{2x – 1}\)

Solution:

As we know, the step to solve the given equation by synthetic division method, we can write;

Synthetic Division Example 4

Synthetic Division Example 4

Example 5:

Divide : \(\frac{x^{3} – 5x^{2} +3x + 7}{x – 3}\)

Solution:

Solving the given expression, by step by step method, we get;

Synthetic Division Example 5

Synthetic Division Example 5

Articles Related to Division by Synthetic Method

Frequently Asked Questions

What is the Main Use of Synthetic Division?

Synthetic division is mainly used to find the zeroes of roots of polynomials.

When Can You Use Synthetic Division?

Synthetic division is used when a polynomial is to be divided by a linear expression and the leading coefficient (first number) must be a 1. For example, any polynomial equation of any degree can be divided by x + 1 but not by x2+1

Why is Synthetic Division Important?

Synthetic division is useful to divide polynomials in an easy and simple way as it breaks down complex equations into smaller and easier equations.

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