Divide Using Synthetic Division (x^8-1)/(x+2)

$\frac{x^{8} - 1}{x + 2}$

Step 1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0								\xa0

Step 2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0								\xa0
	$1$

Step 3

Multiply the newest entry in the result $(1)$ by the divisor $(- 2)$ and place the result of $(- 2)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$							\xa0
	$1$

Step 4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$							\xa0
	$1$	$- 2$

Step 5

Multiply the newest entry in the result $(- 2)$ by the divisor $(- 2)$ and place the result of $(4)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$						\xa0
	$1$	$- 2$

Step 6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$						\xa0
	$1$	$- 2$	$4$

Step 7

Multiply the newest entry in the result $(4)$ by the divisor $(- 2)$ and place the result of $(- 8)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$					\xa0
	$1$	$- 2$	$4$

Step 8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$					\xa0
	$1$	$- 2$	$4$	$- 8$

Step 9

Multiply the newest entry in the result $(- 8)$ by the divisor $(- 2)$ and place the result of $(16)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$	$16$				\xa0
	$1$	$- 2$	$4$	$- 8$

Step 10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$	$16$				\xa0
	$1$	$- 2$	$4$	$- 8$	$16$

Step 11

Multiply the newest entry in the result $(16)$ by the divisor $(- 2)$ and place the result of $(- 32)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$	$16$	$- 32$			\xa0
	$1$	$- 2$	$4$	$- 8$	$16$

Step 12

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$	$16$	$- 32$			\xa0
	$1$	$- 2$	$4$	$- 8$	$16$	$- 32$

Step 13

Multiply the newest entry in the result $(- 32)$ by the divisor $(- 2)$ and place the result of $(64)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$	$16$	$- 32$	$64$		\xa0
	$1$	$- 2$	$4$	$- 8$	$16$	$- 32$

Step 14

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$	$16$	$- 32$	$64$		\xa0
	$1$	$- 2$	$4$	$- 8$	$16$	$- 32$	$64$

Step 15

Multiply the newest entry in the result $(64)$ by the divisor $(- 2)$ and place the result of $(- 128)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$	$16$	$- 32$	$64$	$- 128$	\xa0
	$1$	$- 2$	$4$	$- 8$	$16$	$- 32$	$64$

Step 16

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$	$16$	$- 32$	$64$	$- 128$	\xa0
	$1$	$- 2$	$4$	$- 8$	$16$	$- 32$	$64$	$- 128$

Step 17

Multiply the newest entry in the result $(- 128)$ by the divisor $(- 2)$ and place the result of $(256)$ under the next term in the dividend $(- 1)$ .

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$	$16$	$- 32$	$64$	$- 128$	$256$
	$1$	$- 2$	$4$	$- 8$	$16$	$- 32$	$64$	$- 128$

Step 18

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
	\xa0	$- 2$	$4$	$- 8$	$16$	$- 32$	$64$	$- 128$	$256$
	$1$	$- 2$	$4$	$- 8$	$16$	$- 32$	$64$	$- 128$	$255$

Step 19

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{7} + - 2 x^{6} + 4 x^{5} + - 8 x^{4} + 16 x^{3} + - 32 x^{2} + (64) x - 128 + \frac{255}{x + 2}$

Step 20

Simplify the quotient polynomial.

$x^{7} - 2 x^{6} + 4 x^{5} - 8 x^{4} + 16 x^{3} - 32 x^{2} + 64 x - 128 + \frac{255}{x + 2}$

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