Examples

x^{3} - 3 x^{2} - 2 x + 6

$x^{3} - 3 x^{2} - 2 x + 6$ ,

x - 4

$x - 4$

Step 1

Divide

\frac{x^{3} - 3 x^{2} - 2 x + 6}{x - 4}

$\frac{x^{3} - 3 x^{2} - 2 x + 6}{x - 4}$ using synthetic division and check if the remainder is equal to

0

$0$ . If the remainder is equal to

0

$0$ , it means that

x - 4

$x - 4$ is a factor for

x^{3} - 3 x^{2} - 2 x + 6

$x^{3} - 3 x^{2} - 2 x + 6$ . If the remainder is not equal to

0

$0$ , it means that

x - 4

$x - 4$ is not a factor for

x^{3} - 3 x^{2} - 2 x + 6

$x^{3} - 3 x^{2} - 2 x + 6$ .

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Step 1.1

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

$4$ $4$	$1$ $1$	$- 3$ $- 3$	$- 2$ $- 2$	$6$ $6$

Step 1.2

The first number in the dividend

(1)

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

$4$ $4$	$1$ $1$	$- 3$ $- 3$	$- 2$ $- 2$	$6$ $6$

	$1$ $1$

Step 1.3

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(4)

$(4)$ and place the result of

(4)

$(4)$ under the next term in the dividend

(- 3)

$(- 3)$ .

$4$ $4$	$1$ $1$	$- 3$ $- 3$	$- 2$ $- 2$	$6$ $6$
		$4$ $4$
	$1$ $1$

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.

$4$ $4$	$1$ $1$	$- 3$ $- 3$	$- 2$ $- 2$	$6$ $6$
		$4$ $4$
	$1$ $1$	$1$ $1$

Step 1.5

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(4)

$(4)$ and place the result of

(4)

$(4)$ under the next term in the dividend

(- 2)

$(- 2)$ .

$4$ $4$	$1$ $1$	$- 3$ $- 3$	$- 2$ $- 2$	$6$ $6$
		$4$ $4$	$4$ $4$
	$1$ $1$	$1$ $1$

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.

$4$ $4$	$1$ $1$	$- 3$ $- 3$	$- 2$ $- 2$	$6$ $6$
		$4$ $4$	$4$ $4$
	$1$ $1$	$1$ $1$	$2$ $2$

Step 1.7

Multiply the newest entry in the result

(2)

$(2)$ by the divisor

(4)

$(4)$ and place the result of

(8)

$(8)$ under the next term in the dividend

(6)

$(6)$ .

$4$ $4$	$1$ $1$	$- 3$ $- 3$	$- 2$ $- 2$	$6$ $6$
		$4$ $4$	$4$ $4$	$8$ $8$
	$1$ $1$	$1$ $1$	$2$ $2$

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.

$4$ $4$	$1$ $1$	$- 3$ $- 3$	$- 2$ $- 2$	$6$ $6$
		$4$ $4$	$4$ $4$	$8$ $8$
	$1$ $1$	$1$ $1$	$2$ $2$	$14$ $14$

Step 1.9

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

1 x^{2} + 1 x + 2 + \frac{14}{x - 4}

$1 x^{2} + 1 x + 2 + \frac{14}{x - 4}$

Step 1.10

Simplify the quotient polynomial.

x^{2} + x + 2 + \frac{14}{x - 4}

$x^{2} + x + 2 + \frac{14}{x - 4}$

x^{2} + x + 2 + \frac{14}{x - 4}

$x^{2} + x + 2 + \frac{14}{x - 4}$

Step 2

The remainder from dividing

\frac{x^{3} - 3 x^{2} - 2 x + 6}{x - 4}

$\frac{x^{3} - 3 x^{2} - 2 x + 6}{x - 4}$ is

14

$14$ , which is not equal to

0

$0$ . The remainder is not equal to

0

$0$ means that

x - 4

$x - 4$ is not a factor for

x^{3} - 3 x^{2} - 2 x + 6

$x^{3} - 3 x^{2} - 2 x + 6$ .

x - 4

$x - 4$ is not a factor for

x^{3} - 3 x^{2} - 2 x + 6

$x^{3} - 3 x^{2} - 2 x + 6$

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