Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.(note: remember if a polynomial goes from to there is a zero coefficient for . This is simply because really looks like
5
|
2
-13
0
75
2
0
-50
|
Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 2)
5
|
2
-13
0
75
2
0
-50
|
2
Multiply 5 by 2 and place the product (which is 10) right underneath the second coefficient (which is -13)
5
|
2
-13
0
75
2
0
-50
|
10
2
Add 10 and -13 to get -3. Place the sum right underneath 10.
5
|
2
-13
0
75
2
0
-50
|
10
2
-3
Multiply 5 by -3 and place the product (which is -15) right underneath the third coefficient (which is 0)
5
|
2
-13
0
75
2
0
-50
|
10
-15
2
-3
Add -15 and 0 to get -15. Place the sum right underneath -15.
5
|
2
-13
0
75
2
0
-50
|
10
-15
2
-3
-15
Multiply 5 by -15 and place the product (which is -75) right underneath the fourth coefficient (which is 75)
5
|
2
-13
0
75
2
0
-50
|
10
-15
-75
2
-3
-15
Add -75 and 75 to get 0. Place the sum right underneath -75.
5
|
2
-13
0
75
2
0
-50
|
10
-15
-75
2
-3
-15
0
Multiply 5 by 0 and place the product (which is 0) right underneath the fifth coefficient (which is 2)
5
|
2
-13
0
75
2
0
-50
|
10
-15
-75
0
2
-3
-15
0
Add 0 and 2 to get 2. Place the sum right underneath 0.
5
|
2
-13
0
75
2
0
-50
|
10
-15
-75
0
2
-3
-15
0
2
Multiply 5 by 2 and place the product (which is 10) right underneath the sixth coefficient (which is 0)
5
|
2
-13
0
75
2
0
-50
|
10
-15
-75
0
10
2
-3
-15
0
2
Add 10 and 0 to get 10. Place the sum right underneath 10.
5
|
2
-13
0
75
2
0
-50
|
10
-15
-75
0
10
2
-3
-15
0
2
10
Multiply 5 by 10 and place the product (which is 50) right underneath the seventh coefficient (which is -50)
5
|
2
-13
0
75
2
0
-50
|
10
-15
-75
0
10
50
2
-3
-15
0
2
10
Add 50 and -50 to get 0. Place the sum right underneath 50.
5
|
2
-13
0
75
2
0
-50
|
10
-15
-75
0
10
50
2
-3
-15
0
2
10
0
Since the last column adds to zero, we have a remainder of zero. This means is a factor of
Now lets look at the bottom row of coefficients:
The first 6 coefficients (2,-3,-15,0,2,10) form the quotient
(