Find the Factors Using the Factor Theorem 3x^3-16x^2+23x-6 ; x-2

$3 x^{3} - 16 x^{2} + 23 x - 6$ ; $x - 2$

Divide $\frac{3 x^{3} - 16 x^{2} + 23 x - 6}{x - 2}$ 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 $3 x^{3} - 16 x^{2} + 23 x - 6$ . If the remainder is not equal to $0$ , it means that $x - 2$ is not a factor for $3 x^{3} - 16 x^{2} + 23 x - 6$ .

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

$2$	$3$	$- 16$	$23$	$- 6$
	\xa0			\xa0

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

$2$	$3$	$- 16$	$23$	$- 6$
	\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 $(- 16)$ .

$2$	$3$	$- 16$	$23$	$- 6$
	\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.

$2$	$3$	$- 16$	$23$	$- 6$
	\xa0	$6$		\xa0
	$3$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(2)$ and place the result of $(- 20)$ under the next term in the dividend $(23)$ .

$2$	$3$	$- 16$	$23$	$- 6$
	\xa0	$6$	$- 20$	\xa0
	$3$	$- 10$

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

$2$	$3$	$- 16$	$23$	$- 6$
	\xa0	$6$	$- 20$	\xa0
	$3$	$- 10$	$3$

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 $(- 6)$ .

$2$	$3$	$- 16$	$23$	$- 6$
	\xa0	$6$	$- 20$	$6$
	$3$	$- 10$	$3$

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

$2$	$3$	$- 16$	$23$	$- 6$
	\xa0	$6$	$- 20$	$6$
	$3$	$- 10$	$3$	$0$

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

$3 x^{2} + - 10 x + 3$

Simplify the quotient polynomial.

$3 x^{2} - 10 x + 3$

The remainder from dividing $\frac{3 x^{3} - 16 x^{2} + 23 x - 6}{x - 2}$ is $0$ , which means that $x - 2$ is a factor for $3 x^{3} - 16 x^{2} + 23 x - 6$ .

$x - 2$ is a factor for $3 x^{3} - 16 x^{2} + 23 x - 6$

Find all the possible roots for $3 x^{2} - 10 x + 3$ .

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 3$

$q = \pm 1, \pm 3$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm \frac{1}{3}, \pm 3$

Set up the next division to determine if $x - \frac{1}{3}$ is a factor of the polynomial $3 x^{2} - 10 x + 3$ .

$\frac{3 x^{2} - 10 x + 3}{x - \frac{1}{3}}$

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x - \frac{1}{3}$ divides evenly into $3 x^{2} - 10 x + 3$ , $x - \frac{1}{3}$ is a factor of the polynomial and there is a remaining polynomial of $3 x - 9$ .

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

$\frac{1}{3}$	$3$	$- 10$	$3$
	\xa0		\xa0

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

$\frac{1}{3}$	$3$	$- 10$	$3$
	\xa0		\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(\frac{1}{3})$ and place the result of $(1)$ under the next term in the dividend $(- 10)$ .

$\frac{1}{3}$	$3$	$- 10$	$3$
	\xa0	$1$	\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.

$\frac{1}{3}$	$3$	$- 10$	$3$
	\xa0	$1$	\xa0
	$3$	$- 9$

Multiply the newest entry in the result $(- 9)$ by the divisor $(\frac{1}{3})$ and place the result of $(- 3)$ under the next term in the dividend $(3)$ .

$\frac{1}{3}$	$3$	$- 10$	$3$
	\xa0	$1$	$- 3$
	$3$	$- 9$

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

$\frac{1}{3}$	$3$	$- 10$	$3$
	\xa0	$1$	$- 3$
	$3$	$- 9$	$0$

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

$(3) x - 9$

Simplify the quotient polynomial.

$3 x - 9$

The final factor is the only factor left over from the synthetic division.

$3 x - 9$

The factored polynomial is $(x - 2) (x - \frac{1}{3}) (3 x - 9)$ .

$(x - 2) (x - \frac{1}{3}) (3 x - 9)$

Find the Factors Using the Factor Theorem 3x^3-16x^2+23x-6 ; x-2

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