Find the Factors Using the Factor Theorem 169x^3-1014x^2-49x+294 ; x-6

$169 x^{3} - 1014 x^{2} - 49 x + 294$ ; $x - 6$

Divide $\frac{169 x^{3} - 1014 x^{2} - 49 x + 294}{x - 6}$ 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 $169 x^{3} - 1014 x^{2} - 49 x + 294$ . If the remainder is not equal to $0$ , it means that $x - 6$ is not a factor for $169 x^{3} - 1014 x^{2} - 49 x + 294$ .

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

$6$	$169$	$- 1014$	$- 49$	$294$
	\xa0			\xa0

The first number in the dividend $(169)$ is put into the first position of the result area (below the horizontal line).

$6$	$169$	$- 1014$	$- 49$	$294$
	\xa0			\xa0
	$169$

Multiply the newest entry in the result $(169)$ by the divisor $(6)$ and place the result of $(1014)$ under the next term in the dividend $(- 1014)$ .

$6$	$169$	$- 1014$	$- 49$	$294$
	\xa0	$1014$		\xa0
	$169$

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

$6$	$169$	$- 1014$	$- 49$	$294$
	\xa0	$1014$		\xa0
	$169$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(6)$ and place the result of $(0)$ under the next term in the dividend $(- 49)$ .

$6$	$169$	$- 1014$	$- 49$	$294$
	\xa0	$1014$	$0$	\xa0
	$169$	$0$

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

$6$	$169$	$- 1014$	$- 49$	$294$
	\xa0	$1014$	$0$	\xa0
	$169$	$0$	$- 49$

Multiply the newest entry in the result $(- 49)$ by the divisor $(6)$ and place the result of $(- 294)$ under the next term in the dividend $(294)$ .

$6$	$169$	$- 1014$	$- 49$	$294$
	\xa0	$1014$	$0$	$- 294$
	$169$	$0$	$- 49$

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

$6$	$169$	$- 1014$	$- 49$	$294$
	\xa0	$1014$	$0$	$- 294$
	$169$	$0$	$- 49$	$0$

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

$169 x^{2} + 0 x - 49$

Simplify the quotient polynomial.

$169 x^{2} - 49$

The remainder from dividing $\frac{169 x^{3} - 1014 x^{2} - 49 x + 294}{x - 6}$ is $0$ , which means that $x - 6$ is a factor for $169 x^{3} - 1014 x^{2} - 49 x + 294$ .

$x - 6$ is a factor for $169 x^{3} - 1014 x^{2} - 49 x + 294$

Find all the possible roots for $169 x^{2} - 49$ .

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.

$p = \pm 1, \pm 7, \pm 49$

$q = \pm 1, \pm 13, \pm 169$

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

$\pm 1, \pm \frac{1}{13}, \pm \frac{1}{169}, \pm 7, \pm \frac{7}{13}, \pm \frac{7}{169}, \pm 49, \pm \frac{49}{13}, \pm \frac{49}{169}$

Set up the next division to determine if $x - \frac{7}{13}$ is a factor of the polynomial $169 x^{2} - 49$ .

$\frac{169 x^{2} - 49}{x - \frac{7}{13}}$

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x - \frac{7}{13}$ divides evenly into $169 x^{2} - 49$ , $x - \frac{7}{13}$ is a factor of the polynomial and there is a remaining polynomial of $169 x + 91$ .

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

$\frac{7}{13}$	$169$	$0$	$- 49$
	\xa0		\xa0

The first number in the dividend $(169)$ is put into the first position of the result area (below the horizontal line).

$\frac{7}{13}$	$169$	$0$	$- 49$
	\xa0		\xa0
	$169$

Multiply the newest entry in the result $(169)$ by the divisor $(\frac{7}{13})$ and place the result of $(91)$ under the next term in the dividend $(0)$ .

$\frac{7}{13}$	$169$	$0$	$- 49$
	\xa0	$91$	\xa0
	$169$

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

$\frac{7}{13}$	$169$	$0$	$- 49$
	\xa0	$91$	\xa0
	$169$	$91$

Multiply the newest entry in the result $(91)$ by the divisor $(\frac{7}{13})$ and place the result of $(49)$ under the next term in the dividend $(- 49)$ .

$\frac{7}{13}$	$169$	$0$	$- 49$
	\xa0	$91$	$49$
	$169$	$91$

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

$\frac{7}{13}$	$169$	$0$	$- 49$
	\xa0	$91$	$49$
	$169$	$91$	$0$

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

$(169) x + 91$

Simplify the quotient polynomial.

$169 x + 91$

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

$169 x + 91$

The factored polynomial is $(x - 6) (x - \frac{7}{13}) (169 x + 91)$ .

$(x - 6) (x - \frac{7}{13}) (169 x + 91)$

Find the Factors Using the Factor Theorem 169x^3-1014x^2-49x+294 ; x-6

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