Determine if the Expression is a Factor Using Synthetic Division

$2 x^{3} + 3 x^{2} - 5 x + 3$ , $x + 1$

Step 1

Divide $\frac{2 x^{3} + 3 x^{2} - 5 x + 3}{x + 1}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x + 1$ is a factor for $2 x^{3} + 3 x^{2} - 5 x + 3$ . If the remainder is not equal to $0$ , it means that $x + 1$ is not a factor for $2 x^{3} + 3 x^{2} - 5 x + 3$ .

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Step 1.1

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

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

Step 1.2

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

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

	$2$

Step 1.3

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 $(3)$ .

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

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.

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

Step 1.5

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 $(- 5)$ .

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

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.

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

Step 1.7

Multiply the newest entry in the result $(- 6)$ by the divisor $(- 1)$ and place the result of $(6)$ under the next term in the dividend $(3)$ .

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

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.

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

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.

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

Step 1.10

Simplify the quotient polynomial.

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

Step 2

The remainder from dividing $\frac{2 x^{3} + 3 x^{2} - 5 x + 3}{x + 1}$ is $9$ , which is not equal to $0$ . The remainder is not equal to $0$ means that $x + 1$ is not a factor for $2 x^{3} + 3 x^{2} - 5 x + 3$ .

$x + 1$ is not a factor for $2 x^{3} + 3 x^{2} - 5 x + 3$

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