Determine if the Expression is a Factor Using Synthetic Division x-2 , 3x^4-6x^3-6x+12

$x - 2$ , $3 x^{4} - 6 x^{3} - 6 x + 12$

Step 1

Divide $\frac{3 x^{4} - 6 x^{3} - 6 x + 12}{x - 2}$ 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 $3 x^{4} - 6 x^{3} - 6 x + 12$ . If the remainder is not equal to $0$ , it means that $x - 2$ is not a factor for $3 x^{4} - 6 x^{3} - 6 x + 12$ .

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

$2$	$3$	$- 6$	$0$	$- 6$	$12$
	\xa0				\xa0

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$2$	$3$	$- 6$	$0$	$- 6$	$12$
	\xa0				\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(2)$ and place the result of $(6)$ under the next term in the dividend $(- 6)$ .

$2$	$3$	$- 6$	$0$	$- 6$	$12$
	\xa0	$6$			\xa0
	$3$

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

$2$	$3$	$- 6$	$0$	$- 6$	$12$
	\xa0	$6$			\xa0
	$3$	$0$

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 $(0)$ .

$2$	$3$	$- 6$	$0$	$- 6$	$12$
	\xa0	$6$	$0$		\xa0
	$3$	$0$

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

$2$	$3$	$- 6$	$0$	$- 6$	$12$
	\xa0	$6$	$0$		\xa0
	$3$	$0$	$0$

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 $(- 6)$ .

$2$	$3$	$- 6$	$0$	$- 6$	$12$
	\xa0	$6$	$0$	$0$	\xa0
	$3$	$0$	$0$

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

$2$	$3$	$- 6$	$0$	$- 6$	$12$
	\xa0	$6$	$0$	$0$	\xa0
	$3$	$0$	$0$	$- 6$

Multiply the newest entry in the result $(- 6)$ by the divisor $(2)$ and place the result of $(- 12)$ under the next term in the dividend $(12)$ .

$2$	$3$	$- 6$	$0$	$- 6$	$12$
	\xa0	$6$	$0$	$0$	$- 12$
	$3$	$0$	$0$	$- 6$

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

$2$	$3$	$- 6$	$0$	$- 6$	$12$
	\xa0	$6$	$0$	$0$	$- 12$
	$3$	$0$	$0$	$- 6$	$0$

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

$3 x^{3} + 0 x^{2} + (0) x - 6$

Simplify the quotient polynomial.

$3 x^{3} - 6$

Step 2

The remainder from dividing $\frac{3 x^{4} - 6 x^{3} - 6 x + 12}{x - 2}$ is $0$ , which means that $x - 2$ is a factor for $3 x^{4} - 6 x^{3} - 6 x + 12$ .

$x - 2$ is a factor for $3 x^{4} - 6 x^{3} - 6 x + 12$

Determine if the Expression is a Factor Using Synthetic Division x-2 , 3x^4-6x^3-6x+12

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