Synthetic division

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

Synthetic Division Formula

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

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 – 1/4  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

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 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

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 consist of linear equation, so we can easily apply the synthetic division method here.

Follow step by step method as given below;

Synthetic Division Examples

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

Solution: Following the same steps as per previous examples.

Synthetic Division Problem

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 Formula

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

Solution: Solving the given experssion, by step by step method, we get;

Synthetic Division Steps

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