SOLUTION: Factor the polynomial below completely, given the bionomial folowing it is a factor of thepoynomial x^3-4x^2+x+6, x+1

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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 -1


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.

-1	|	1	-4	1	6
	|

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

-1	|	1	-4	1	6
	|
		1

Multiply -1 by 1 and place the product (which is -1) right underneath the second coefficient (which is -4)

-1	|	1	-4	1	6
	|		-1
		1

Add -1 and -4 to get -5. Place the sum right underneath -1.

-1	|	1	-4	1	6
	|		-1
		1	-5

Multiply -1 by -5 and place the product (which is 5) right underneath the third coefficient (which is 1)

-1	|	1	-4	1	6
	|		-1	5
		1	-5

Add 5 and 1 to get 6. Place the sum right underneath 5.

-1	|	1	-4	1	6
	|		-1	5
		1	-5	6

Multiply -1 by 6 and place the product (which is -6) right underneath the fourth coefficient (which is 6)

-1	|	1	-4	1	6
	|		-1	5	-6
		1	-5	6

Add -6 and 6 to get 0. Place the sum right underneath -6.

-1	|	1	-4	1	6
	|		-1	5	-6
		1	-5	6	0

Now lets look at the bottom row of coefficients:

The first 3 coefficients (1,-5,6) form the quotient




So

Basically factors to

Now lets break down further




Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .


Now multiply the first coefficient by the last term to get .


Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?


To find these two numbers, we need to list all of the factors of (the previous product).


Factors of :
1,2,3,6
-1,-2,-3,-6


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to .
1*6
2*3
(-1)*(-6)
(-2)*(-3)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number	Second Number	Sum
1	6	1+6=7
2	3	2+3=5
-1	-6	-1+(-6)=-7
-2	-3	-2+(-3)=-5

From the table, we can see that the two numbers and add to (the middle coefficient).


So the two numbers and both multiply to and add to


Now replace the middle term with . Remember, and add to . So this shows us that .


Replace the second term with .


Group the terms into two pairs.


Factor out the GCF from the first group.


Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


Combine like terms. Or factor out the common term



So factors to .


So now becomes

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Answer:


So completely factors to