SOLUTION: please me find the answer by useing the Factor Theorem to determine whether (x- 3) is a factor of f(x) = x^4 + 12x^3 + 6x + 27

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Start with the given polynomial

First lets find our test zero:

Set the denominator equal to zero

Solve for x.

so our test zero is 3


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

3	|	1	12	0	6	27
	|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)

3	|	1	12	0	6	27
	|
		1

Multiply 3 by 1 and place the product (which is 3) right underneath the second coefficient (which is 12)

3	|	1	12	0	6	27
	|		3
		1

Add 3 and 12 to get 15. Place the sum right underneath 3.

3	|	1	12	0	6	27
	|		3
		1	15

Multiply 3 by 15 and place the product (which is 45) right underneath the third coefficient (which is 0)

3	|	1	12	0	6	27
	|		3	45
		1	15

Add 45 and 0 to get 45. Place the sum right underneath 45.

3	|	1	12	0	6	27
	|		3	45
		1	15	45

Multiply 3 by 45 and place the product (which is 135) right underneath the fourth coefficient (which is 6)

3	|	1	12	0	6	27
	|		3	45	135
		1	15	45

Add 135 and 6 to get 141. Place the sum right underneath 135.

3	|	1	12	0	6	27
	|		3	45	135
		1	15	45	141

Multiply 3 by 141 and place the product (which is 423) right underneath the fifth coefficient (which is 27)

3	|	1	12	0	6	27
	|		3	45	135	423
		1	15	45	141

Add 423 and 27 to get 450. Place the sum right underneath 423.

3	|	1	12	0	6	27
	|		3	45	135	423
		1	15	45	141	450

Since the last column adds to 450, we have a remainder of 450. This means is not a factor of
Now lets look at the bottom row of coefficients:

The first 4 coefficients (1,15,45,141) form the quotient



and the last coefficient 450, is the remainder, which is placed over like this





Putting this altogether, we get:



So

which looks like this in remainder form:
remainder 450


You can use this online polynomial division calculator to check your work