Algebra Examples

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

Step 1

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

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$

Step 2

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

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$

	$1$

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 $(0)$ .

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
		$- 2$
	$1$

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.

$- 2$	$1$	$0$	$1$	$0$	$- 10$	$0$	$0$	$12$
		$- 2$
	$1$	$- 2$

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 $(1)$ .

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

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.

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

Step 7

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 $(0)$ .

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

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.

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

Step 9

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 $(- 10)$ .

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

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.

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

Step 11

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 $(0)$ .

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

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.

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

Step 13

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 $(0)$ .

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

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.

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

Step 15

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 $(12)$ .

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

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.

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

Step 17

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

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

Step 18

Simplify the quotient polynomial.

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

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$