# 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;

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;

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

Solution: Following the same steps as per previous examples.

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;

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

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

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