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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 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 terms
Diagonal pattern: MULTIPLY terms
Example
Divide:
 Step 1:

x 2  5x  6
x 1
 Write
the coefficients of the dividend in a
upside-down division symbol.
1
5
6
x  5x  6
x 1
2
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 1
5
6
Example

Step 3:
 Carry
x 2  5x  6
x 1
down the first coefficient.
1 1
1
5
6
Example

Step 4:
x 2  5x  6
x 1
 Multiply
the zero by this number. Write the
product under the next coefficient.
1 1
1
5
1
6
Example

Step 5:
x 2  5x  6
x 1
 Add.
1 1
5
1
1
6
6
Example
x 2  5x  6
x 1
 Step 6 etc.:
 Repeat
as necessary
1 1
1
5
1
6
6
6 12
step 7
The numbers at the bottom represent the
coefficients of the answer. The new
polynomial will be one degree less than
the original.
x 2  5x  6

x 1
1 1 5 6
1 6
12
x6
1 6 12
x 1
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:

2x2 + 3x + 4
x-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.
1 2
3
4
Example
2x2 + 3x + 4
x-1

Step 3:
 Carry
down the first coefficient.
1 2
2
3
4
Example
2x2 + 3x + 4
x-1

Step 4:
 Multiply
the zero by this number. Write the
product under the next coefficient.
1 2
2
3
2
4
Example
2x2 + 3x + 4
x-1

Step 5:
 Add.
1 2
3
2
2
5
4
Example
2x2 + 3x + 4
x-1

Step 6 etc.:
 Repeat
as necessary
1 2
3
2
2
5
4
5
9
step 7
The numbers at the bottom represent the
coefficients of the answer. The new
polynomial will be one degree less than
the original.
x 2  5x  6

x 1
1 2 3 4
2 5
2 5 9
****Lab Application, practice