SOLUTION: solve using synthetic division (x^4+6x³+6x²)÷(x+5)

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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 -5


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	|	1	6	6	0	0
	|

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

-5	|	1	6	6	0	0
	|
		1

Multiply -5 by 1 and place the product (which is -5) right underneath the second coefficient (which is 6)

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

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

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

Multiply -5 by 1 and place the product (which is -5) right underneath the third coefficient (which is 6)

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

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

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

Multiply -5 by 1 and place the product (which is -5) right underneath the fourth coefficient (which is 0)

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

Add -5 and 0 to get -5. Place the sum right underneath -5.

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

Multiply -5 by -5 and place the product (which is 25) right underneath the fifth coefficient (which is 0)

-5	|	1	6	6	0	0
	|		-5	-5	-5	25
		1	1	1	-5

Add 25 and 0 to get 25. Place the sum right underneath 25.

-5	|	1	6	6	0	0
	|		-5	-5	-5	25
		1	1	1	-5	25

Since the last column adds to 25, we have a remainder of 25. This means is not a factor of
Now lets look at the bottom row of coefficients:

The first 4 coefficients (1,1,1,-5) form the quotient



and the last coefficient 25, is the remainder, which is placed over like this





Putting this altogether, we get:



So

which looks like this in remainder form:
remainder 25


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