Precalculus Examples

$\frac{2 x^{5} - x^{4} + 3 x^{2} - x + 5}{x - 1}$

Step 1

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

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$

Step 2

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$

	$2$

Step 3

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

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$
		$2$
	$2$

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.

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$
		$2$
	$2$	$1$

Step 5

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

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$
		$2$	$1$
	$2$	$1$

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.

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$
		$2$	$1$
	$2$	$1$	$1$

Step 7

Multiply the newest entry in the result $(1)$ by the divisor $(1)$ and place the result of $(1)$ under the next term in the dividend $(3)$ .

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$
		$2$	$1$	$1$
	$2$	$1$	$1$

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.

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$
		$2$	$1$	$1$
	$2$	$1$	$1$	$4$

Step 9

Multiply the newest entry in the result $(4)$ by the divisor $(1)$ and place the result of $(4)$ under the next term in the dividend $(- 1)$ .

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$
		$2$	$1$	$1$	$4$
	$2$	$1$	$1$	$4$

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.

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$
		$2$	$1$	$1$	$4$
	$2$	$1$	$1$	$4$	$3$

Step 11

Multiply the newest entry in the result $(3)$ by the divisor $(1)$ and place the result of $(3)$ under the next term in the dividend $(5)$ .

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$
		$2$	$1$	$1$	$4$	$3$
	$2$	$1$	$1$	$4$	$3$

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.

$1$	$2$	$- 1$	$0$	$3$	$- 1$	$5$
		$2$	$1$	$1$	$4$	$3$
	$2$	$1$	$1$	$4$	$3$	$8$

Step 13

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

$2 x^{4} + 1 x^{3} + 1 x^{2} + 4 x + 3 + \frac{8}{x - 1}$

Step 14

Simplify the quotient polynomial.

$2 x^{4} + x^{3} + x^{2} + 4 x + 3 + \frac{8}{x - 1}$

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