Find the Factors Using the Factor Theorem x-1/6 , 6x^4-x^3+6x-1

$x - \frac{1}{6}$ , $6 x^{4} - x^{3} + 6 x - 1$

Step 1

Divide $\frac{6 x^{4} - x^{3} + 6 x - 1}{x - \frac{1}{6}}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x - \frac{1}{6}$ is a factor for $6 x^{4} - x^{3} + 6 x - 1$ . If the remainder is not equal to $0$ , it means that $x - \frac{1}{6}$ is not a factor for $6 x^{4} - x^{3} + 6 x - 1$ .

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

$\frac{1}{6}$	$6$	$- 1$	$0$	$6$	$- 1$
	\xa0				\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$\frac{1}{6}$	$6$	$- 1$	$0$	$6$	$- 1$
	\xa0				\xa0
	$6$

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

$\frac{1}{6}$	$6$	$- 1$	$0$	$6$	$- 1$
	\xa0	$1$			\xa0
	$6$

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

$\frac{1}{6}$	$6$	$- 1$	$0$	$6$	$- 1$
	\xa0	$1$			\xa0
	$6$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(\frac{1}{6})$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$\frac{1}{6}$	$6$	$- 1$	$0$	$6$	$- 1$
	\xa0	$1$	$0$		\xa0
	$6$	$0$

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

$\frac{1}{6}$	$6$	$- 1$	$0$	$6$	$- 1$
	\xa0	$1$	$0$		\xa0
	$6$	$0$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(\frac{1}{6})$ and place the result of $(0)$ under the next term in the dividend $(6)$ .

$\frac{1}{6}$	$6$	$- 1$	$0$	$6$	$- 1$
	\xa0	$1$	$0$	$0$	\xa0
	$6$	$0$	$0$

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

$\frac{1}{6}$	$6$	$- 1$	$0$	$6$	$- 1$
	\xa0	$1$	$0$	$0$	\xa0
	$6$	$0$	$0$	$6$

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

$\frac{1}{6}$	$6$	$- 1$	$0$	$6$	$- 1$
	\xa0	$1$	$0$	$0$	$1$
	$6$	$0$	$0$	$6$

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

$\frac{1}{6}$	$6$	$- 1$	$0$	$6$	$- 1$
	\xa0	$1$	$0$	$0$	$1$
	$6$	$0$	$0$	$6$	$0$

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

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

Simplify the quotient polynomial.

$6 x^{3} + 6$

Step 2

The remainder from dividing $\frac{6 x^{4} - x^{3} + 6 x - 1}{x - \frac{1}{6}}$ is $0$ , which means that $x - \frac{1}{6}$ is a factor for $6 x^{4} - x^{3} + 6 x - 1$ .

$x - \frac{1}{6}$ is a factor for $6 x^{4} - x^{3} + 6 x - 1$

Step 3

Find all the possible roots for $6 x^{3} + 6$ .

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.

$p = \pm 1, \pm 2, \pm 3, \pm 6$

$q = \pm 1, \pm 2, \pm 3, \pm 6$

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm \frac{3}{2}, \pm 6$

Step 4

Set up the next division to determine if $x + 1$ is a factor of the polynomial $6 x^{3} + 6$ .

$\frac{6 x^{3} + 6}{x + 1}$

Step 5

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x + 1$ divides evenly into $6 x^{3} + 6$ , $x + 1$ is a factor of the polynomial and there is a remaining polynomial of $6 x^{2} - 6 x + 6$ .

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

$- 1$	$6$	$0$	$0$	$6$
	\xa0			\xa0

The first number in the dividend $(6)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$6$	$0$	$0$	$6$
	\xa0			\xa0
	$6$

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 $(0)$ .

$- 1$	$6$	$0$	$0$	$6$
	\xa0	$- 6$		\xa0
	$6$

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

$- 1$	$6$	$0$	$0$	$6$
	\xa0	$- 6$		\xa0
	$6$	$- 6$

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 $(0)$ .

$- 1$	$6$	$0$	$0$	$6$
	\xa0	$- 6$	$6$	\xa0
	$6$	$- 6$

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

$- 1$	$6$	$0$	$0$	$6$
	\xa0	$- 6$	$6$	\xa0
	$6$	$- 6$	$6$

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 $(6)$ .

$- 1$	$6$	$0$	$0$	$6$
	\xa0	$- 6$	$6$	$- 6$
	$6$	$- 6$	$6$

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

$- 1$	$6$	$0$	$0$	$6$
	\xa0	$- 6$	$6$	$- 6$
	$6$	$- 6$	$6$	$0$

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

$6 x^{2} + - 6 x + 6$

Simplify the quotient polynomial.

$6 x^{2} - 6 x + 6$

Step 6

Find all the possible roots for $6 x^{2} - 6 x + 6$ .

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.

$p = \pm 1, \pm 2, \pm 3, \pm 6$

$q = \pm 1, \pm 2, \pm 3, \pm 6$

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

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm \frac{3}{2}, \pm 6$

Step 7

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

$6 x^{2} - 6 x + 6$

Step 8

The factored polynomial is $(x - \frac{1}{6}) (x + 1) (6 x^{2} - 6 x + 6)$ .

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

Find the Factors Using the Factor Theorem x-1/6 , 6x^4-x^3+6x-1

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