Algebra Examples

2 x^{3} + 3 x^{2} - 5 x + 3

$2 x^{3} + 3 x^{2} - 5 x + 3$ ,

x + 1

$x + 1$

Step 1

Divide

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

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

$x + 1$ is a factor for

2 x^{3} + 3 x^{2} - 5 x + 3

$2 x^{3} + 3 x^{2} - 5 x + 3$ . If the remainder is not equal to

0

$0$ , it means that

x + 1

$x + 1$ is not a factor for

2 x^{3} + 3 x^{2} - 5 x + 3

$2 x^{3} + 3 x^{2} - 5 x + 3$ .

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Step 1.1

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

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

Step 1.2

The first number in the dividend

(2)

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

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

	$2$ $2$

Step 1.3

Multiply the newest entry in the result

(2)

$(2)$ by the divisor

(- 1)

$(- 1)$ and place the result of

(- 2)

$(- 2)$ under the next term in the dividend

(3)

$(3)$ .

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

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.

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

Step 1.5

Multiply the newest entry in the result

(1)

$(1)$ by the divisor

(- 1)

$(- 1)$ and place the result of

(- 1)

$(- 1)$ under the next term in the dividend

(- 5)

$(- 5)$ .

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

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.

$- 1$ $- 1$	$2$ $2$	$3$ $3$	$- 5$ $- 5$	$3$ $3$
		$- 2$ $- 2$	$- 1$ $- 1$
	$2$ $2$	$1$ $1$	$- 6$ $- 6$

Step 1.7

Multiply the newest entry in the result

(- 6)

$(- 6)$ by the divisor

(- 1)

$(- 1)$ and place the result of

(6)

$(6)$ under the next term in the dividend

(3)

$(3)$ .

$- 1$ $- 1$	$2$ $2$	$3$ $3$	$- 5$ $- 5$	$3$ $3$
		$- 2$ $- 2$	$- 1$ $- 1$	$6$ $6$
	$2$ $2$	$1$ $1$	$- 6$ $- 6$

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.

$- 1$ $- 1$	$2$ $2$	$3$ $3$	$- 5$ $- 5$	$3$ $3$
		$- 2$ $- 2$	$- 1$ $- 1$	$6$ $6$
	$2$ $2$	$1$ $1$	$- 6$ $- 6$	$9$ $9$

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.

2 x^{2} + 1 x - 6 + \frac{9}{x + 1}

$2 x^{2} + 1 x - 6 + \frac{9}{x + 1}$

Step 1.10

Simplify the quotient polynomial.

2 x^{2} + x - 6 + \frac{9}{x + 1}

$2 x^{2} + x - 6 + \frac{9}{x + 1}$

2 x^{2} + x - 6 + \frac{9}{x + 1}

$2 x^{2} + x - 6 + \frac{9}{x + 1}$

Step 2

The remainder from dividing

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

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

9

$9$ , which is not equal to

0

$0$ . The remainder is not equal to

0

$0$ means that

x + 1

$x + 1$ is not a factor for

2 x^{3} + 3 x^{2} - 5 x + 3

$2 x^{3} + 3 x^{2} - 5 x + 3$ .

x + 1

$x + 1$ is not a factor for

2 x^{3} + 3 x^{2} - 5 x + 3

$2 x^{3} + 3 x^{2} - 5 x + 3$

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