SOLUTION: Find the quotient of x^5 + 32 divided by x+2

Question 89662: Find the quotient of x^5 + 32 divided by x+2
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Lets find the quotient using synthetic division

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 -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 , , , and . This is simply because really looks like

-2	\|	1	0	0	0	0	32
	\|

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

-2	\|	1	0	0	0	0	32
	\|
		1

Multiply -2 by 1 and place the product (which is -2) right underneath the second coefficient (which is 0)

-2	\|	1	0	0	0	0	32
	\|		-2
		1

Add -2 and 0 to get -2. Place the sum right underneath -2.

-2	\|	1	0	0	0	0	32
	\|		-2
		1	-2

Multiply -2 by -2 and place the product (which is 4) right underneath the third coefficient (which is 0)

-2	\|	1	0	0	0	0	32
	\|		-2	4
		1	-2

Add 4 and 0 to get 4. Place the sum right underneath 4.

-2	\|	1	0	0	0	0	32
	\|		-2	4
		1	-2	4

Multiply -2 by 4 and place the product (which is -8) right underneath the fourth coefficient (which is 0)

-2	\|	1	0	0	0	0	32
	\|		-2	4	-8
		1	-2	4

Add -8 and 0 to get -8. Place the sum right underneath -8.

-2	\|	1	0	0	0	0	32
	\|		-2	4	-8
		1	-2	4	-8

Multiply -2 by -8 and place the product (which is 16) right underneath the fifth coefficient (which is 0)

-2	\|	1	0	0	0	0	32
	\|		-2	4	-8	16
		1	-2	4	-8

Add 16 and 0 to get 16. Place the sum right underneath 16.

-2	\|	1	0	0	0	0	32
	\|		-2	4	-8	16
		1	-2	4	-8	16

Multiply -2 by 16 and place the product (which is -32) right underneath the sixth coefficient (which is 32)

-2	\|	1	0	0	0	0	32
	\|		-2	4	-8	16	-32
		1	-2	4	-8	16

Add -32 and 32 to get 0. Place the sum right underneath -32.

-2	\|	1	0	0	0	0	32
	\|		-2	4	-8	16	-32
		1	-2	4	-8	16	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 5 coefficients (1,-2,4,-8,16) form the quotient

So

In other words the quotient is
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