Examples

$x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4$ , $x - 2$

Step 1

Divide $\frac{x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4}{x - 2}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x - 2$ is a factor for $x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4$ . If the remainder is not equal to $0$ , it means that $x - 2$ is not a factor for $x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4$ .

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Step 1.1

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

$2$	$1$	$- 2$	$- 10$	$7$	$4$

Step 1.2

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

$2$	$1$	$- 2$	$- 10$	$7$	$4$

	$1$

Step 1.3

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 $(- 2)$ .

$2$	$1$	$- 2$	$- 10$	$7$	$4$
		$2$
	$1$

Step 1.4

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

$2$	$1$	$- 2$	$- 10$	$7$	$4$
		$2$
	$1$	$0$

Step 1.5

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

$2$	$1$	$- 2$	$- 10$	$7$	$4$
		$2$	$0$
	$1$	$0$

Step 1.6

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

$2$	$1$	$- 2$	$- 10$	$7$	$4$
		$2$	$0$
	$1$	$0$	$- 10$

Step 1.7

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 $(7)$ .

$2$	$1$	$- 2$	$- 10$	$7$	$4$
		$2$	$0$	$- 20$
	$1$	$0$	$- 10$

Step 1.8

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

$2$	$1$	$- 2$	$- 10$	$7$	$4$
		$2$	$0$	$- 20$
	$1$	$0$	$- 10$	$- 13$

Step 1.9

Multiply the newest entry in the result $(- 13)$ by the divisor $(2)$ and place the result of $(- 26)$ under the next term in the dividend $(4)$ .

$2$	$1$	$- 2$	$- 10$	$7$	$4$
		$2$	$0$	$- 20$	$- 26$
	$1$	$0$	$- 10$	$- 13$

Step 1.10

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

$2$	$1$	$- 2$	$- 10$	$7$	$4$
		$2$	$0$	$- 20$	$- 26$
	$1$	$0$	$- 10$	$- 13$	$- 22$

Step 1.11

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

$1 x^{3} + 0 x^{2} + (- 10) x - 13 + \frac{- 22}{x - 2}$

Step 1.12

Simplify the quotient polynomial.

$x^{3} - 10 x - 13 - \frac{22}{x - 2}$

Step 2

The remainder from dividing $\frac{x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4}{x - 2}$ is $- 22$ , which is not equal to $0$ . The remainder is not equal to $0$ means that $x - 2$ is not a factor for $x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4$ .

$x - 2$ is not a factor for $x^{4} - 2 x^{3} - 10 x^{2} + 7 x + 4$

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