Let's simplify this expression using synthetic division
Start with the given expression
First lets find our test zero:
Set the denominator equal to zero
Solve for x.
so our test zero is 6
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.
6
|
1
-13
54
-72
|
Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
6
|
1
-13
54
-72
|
1
Multiply 6 by 1 and place the product (which is 6) right underneath the second coefficient (which is -13)
6
|
1
-13
54
-72
|
6
1
Add 6 and -13 to get -7. Place the sum right underneath 6.
6
|
1
-13
54
-72
|
6
1
-7
Multiply 6 by -7 and place the product (which is -42) right underneath the third coefficient (which is 54)
6
|
1
-13
54
-72
|
6
-42
1
-7
Add -42 and 54 to get 12. Place the sum right underneath -42.
6
|
1
-13
54
-72
|
6
-42
1
-7
12
Multiply 6 by 12 and place the product (which is 72) right underneath the fourth coefficient (which is -72)
6
|
1
-13
54
-72
|
6
-42
72
1
-7
12
Add 72 and -72 to get 0. Place the sum right underneath 72.
6
|
1
-13
54
-72
|
6
-42
72
1
-7
12
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 3 coefficients (1,-7,12) form the quotient
(