Polynomial and Synthetic Division

#3

Now let’s look at another method to divide… Why??? Sometimes it is easier…

Synthetic Division

Synthetic Division is a ‘shortcut’ for polynomial division that only works when dividing by a linear factor (x + b).

It involves the coefficients of the dividend, and the zero of the divisor.

Synthetic DivisionSynthetic Division

The pattern for synthetic division of a cubic polynomial is summarized as follows. (The pattern for higher-degree polynomials is similar.)

most

vertical pattern: ADD termsDiagonal pattern: MULTIPLY terms

Example

Divide: Step 1:

Write the coefficients of the dividend in a upside-down division symbol.

1 5 6

1

652

x

xx

Example Step 2:

Take the zero (or root) from the divisor, and write it on the left, x – 1 = 0 , so the zero is 1.

1 5 61

1

652

x

xx

Example

Step 3:Carry down the first coefficient.

1 5 61

1

1

652

x

xx

Example

Step 4:Multiply the zero by this number. Write the

product under the next coefficient.

1 5 61

1

1

1

652

x

xx

Example

Step 5:Add.

1 5 61

1

1

6

1

652

x

xx

Example

Step 6 etc.:Repeat as necessary

1 5 61

1

1

6

6

12

1

652

x

xx

step 7

The numbers at the bottom represent the coefficients of the answer. The new polynomial will be one degree less than the original.

1 5 61

1

1

6

6

12 1

126

x

x

1

652

x

xx

Using Synthetic Division

Use synthetic division to divide x4 – 10x2 – 2x + 4 by x + 3.

Solution:

You should set up the array as follows. Note that a zero is included for the missing +x3 term in the dividend.

Example

Divide: Step 1:

Write the coefficients of the dividend in a upside-down division symbol.

2 3 4

2x2 + 3x + 4 x - 1

Example Step 2:

Take the zero (or root) from the divisor, and write it on the left, x – 1 = 0 , so the zero is 1.

2 3 41

2x2 + 3x + 4 x - 1

Example

Step 3:Carry down the first coefficient.

2 3 41

2

2x2 + 3x + 4 x - 1

Example

Step 4:Multiply the zero by this number. Write the

product under the next coefficient.

2 3 41

2

2

2x2 + 3x + 4 x - 1

Example

Step 5:Add.

2 3 41

2

2

5

2x2 + 3x + 4 x - 1

Example

Step 6 etc.:Repeat as necessary

2 3 41

2

2

5

5

9

2x2 + 3x + 4 x - 1

step 7

The numbers at the bottom represent the coefficients of the answer. The new polynomial will be one degree less than the original.

2 3 41

2

2

5

5

9

1

652

x

xx

****Lab Application, practice