Determine if the Expression is a Factor Using Synthetic Division x^3+10x^2+19x-30 , x+6

$x^{3} + 10 x^{2} + 19 x - 30$ , $x + 6$

Step 1

Divide $\frac{x^{3} + 10 x^{2} + 19 x - 30}{x + 6}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x + 6$ is a factor for $x^{3} + 10 x^{2} + 19 x - 30$ . If the remainder is not equal to $0$ , it means that $x + 6$ is not a factor for $x^{3} + 10 x^{2} + 19 x - 30$ .

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

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0			\xa0

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

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 6)$ and place the result of $(- 6)$ under the next term in the dividend $(10)$ .

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0	$- 6$		\xa0
	$1$

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

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0	$- 6$		\xa0
	$1$	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- 6)$ and place the result of $(- 24)$ under the next term in the dividend $(19)$ .

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0	$- 6$	$- 24$	\xa0
	$1$	$4$

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

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0	$- 6$	$- 24$	\xa0
	$1$	$4$	$- 5$

Multiply the newest entry in the result $(- 5)$ by the divisor $(- 6)$ and place the result of $(30)$ under the next term in the dividend $(- 30)$ .

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0	$- 6$	$- 24$	$30$
	$1$	$4$	$- 5$

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

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0	$- 6$	$- 24$	$30$
	$1$	$4$	$- 5$	$0$

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

$1 x^{2} + 4 x - 5$

Simplify the quotient polynomial.

$x^{2} + 4 x - 5$

Step 2

The remainder from dividing $\frac{x^{3} + 10 x^{2} + 19 x - 30}{x + 6}$ is $0$ , which means that $x + 6$ is a factor for $x^{3} + 10 x^{2} + 19 x - 30$ .

$x + 6$ is a factor for $x^{3} + 10 x^{2} + 19 x - 30$

Determine if the Expression is a Factor Using Synthetic Division x^3+10x^2+19x-30 , x+6

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