Synthetic Division

Synthetic division is a shortcut for polynomial division when the divisor is of the form x – a. Only coefficients are used when dividing with synthetic division.

Example 1: Divide (2 x – 11 + 3 x³) by ( x – 3).

First, this problem will be done in the traditional manner. Then it will be done by using the synthetic division method.

In the traditional manner,

The answer is

To do the problem using synthetic division, follow this procedure:

1. Write the polynomial being divided in descending order. Then write only its coefficients, using 0 for any missing terms.
   

   
   

2. Write the constant, a, of the divisor, x – a, to the left. In this problem, a = 3 .
   

   
   

3. Bring down the first coefficient as shown.
   

   
   

4. Multiply the first coefficient by a. Then write this product under the second coefficient .
   

   
   

5. Add the second coefficient with the product and write the sum as shown.
   

   
   

6. Continue this process of multiplying and adding until there is a sum for the last column.
   

   
   

   The numbers along the bottom row are the coefficients of the quotients with the powers of x in descending order. The last coefficient is the remainder. The first power is one less than the highest power of the polynomial that was being divided.

The division answer is

Example 2: Divide (5 x⁴ + 6 x³ – 9 x² – 7 x + 6) by ( x + 2) using the synthetic method.

The divisor, x + 2, is the same as x – (–2). Now, the divisor is of the form x – a, with a = –2.

The answer is

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