Divide Using Synthetic Division (x^7+x^5-10x^3+12)/(x+2)

$\frac{x^{7} + x^{5} - 10 x^{3} + 12}{x + 2}$

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

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0							\xa0

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

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0							\xa0
	$1$

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 $(0)$ .

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$						\xa0
	$1$

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

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$						\xa0
	$1$	$- 2$

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 $(1)$ .

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$					\xa0
	$1$	$- 2$

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

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$					\xa0
	$1$	$- 2$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(- 2)$ and place the result of $(- 10)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$	$- 10$				\xa0
	$1$	$- 2$	$5$

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

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$	$- 10$				\xa0
	$1$	$- 2$	$5$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(- 2)$ and place the result of $(20)$ under the next term in the dividend $(- 10)$ .

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$	$- 10$	$20$			\xa0
	$1$	$- 2$	$5$	$- 10$

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

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$	$- 10$	$20$			\xa0
	$1$	$- 2$	$5$	$- 10$	$10$

Multiply the newest entry in the result $(10)$ by the divisor $(- 2)$ and place the result of $(- 20)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$	$- 10$	$20$	$- 20$		\xa0
	$1$	$- 2$	$5$	$- 10$	$10$

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

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$	$- 10$	$20$	$- 20$		\xa0
	$1$	$- 2$	$5$	$- 10$	$10$	$- 20$

Multiply the newest entry in the result $(- 20)$ by the divisor $(- 2)$ and place the result of $(40)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$	$- 10$	$20$	$- 20$	$40$	\xa0
	$1$	$- 2$	$5$	$- 10$	$10$	$- 20$

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

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$	$- 10$	$20$	$- 20$	$40$	\xa0
	$1$	$- 2$	$5$	$- 10$	$10$	$- 20$	$40$

Multiply the newest entry in the result $(40)$ by the divisor $(- 2)$ and place the result of $(- 80)$ under the next term in the dividend $(12)$ .

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$	$- 10$	$20$	$- 20$	$40$	$- 80$
	$1$	$- 2$	$5$	$- 10$	$10$	$- 20$	$40$

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

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
	\xa0	$- 2$	$4$	$- 10$	$20$	$- 20$	$40$	$- 80$
	$1$	$- 2$	$5$	$- 10$	$10$	$- 20$	$40$	$- 68$

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

$1 x^{6} + - 2 x^{5} + 5 x^{4} + - 10 x^{3} + 10 x^{2} + - 20 x + 40 + \frac{- 68}{x + 2}$

Simplify the quotient polynomial.

$x^{6} - 2 x^{5} + 5 x^{4} - 10 x^{3} + 10 x^{2} - 20 x + 40 - \frac{68}{x + 2}$

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