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