In this problem, we can either use polynomial long division, or synthetic division.
Polynomial Long Division:
-3x^2 - 6x - 9
---------------------------
x-2 ) -3x^3 + 0x^2 + 3x - 3
-3x^3 - 6x^2
------------
6x^2 + 3x
6x^2 +12x
---------
-9x - 3
-9x +18
------
-21
So
which can be written as
remainder -21
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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 2
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
Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is -3)
Multiply 2 by -3 and place the product (which is -6) right underneath the second coefficient (which is 0)
Add -6 and 0 to get -6. Place the sum right underneath -6.
Multiply 2 by -6 and place the product (which is -12) right underneath the third coefficient (which is 3)
Add -12 and 3 to get -9. Place the sum right underneath -12.
Multiply 2 by -9 and place the product (which is -18) right underneath the fourth coefficient (which is -3)
Add -18 and -3 to get -21. Place the sum right underneath -18.
| 2 | | | -3 | 0 | 3 | -3 |
| | | | -6 | -12 | -18 | |
| | -3 | -6 | -9 | -21 |
Since the last column adds to -21, we have a remainder of -21. This means is not a factor of
Now lets look at the bottom row of coefficients:
The first 3 coefficients (-3,-6,-9) form the quotient
and the last coefficient -21, is the remainder, which is placed over like this
Putting this altogether, we get:
So
which looks like this in remainder form:
remainder -21
You can use this online polynomial division calculator to check your work