Find the Factors Using the Factor Theorem x^3-8x^2+17x-30 ; x-6

$x^{3} - 8 x^{2} + 17 x - 30$ ; $x - 6$

Step 1

Divide $\frac{x^{3} - 8 x^{2} + 17 x - 30}{x - 6}$ 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 $x^{3} - 8 x^{2} + 17 x - 30$ . If the remainder is not equal to $0$ , it means that $x - 6$ is not a factor for $x^{3} - 8 x^{2} + 17 x - 30$ .

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

$6$	$1$	$- 8$	$17$	$- 30$
	\xa0			\xa0

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

$6$	$1$	$- 8$	$17$	$- 30$
	\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 $(- 8)$ .

$6$	$1$	$- 8$	$17$	$- 30$
	\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.

$6$	$1$	$- 8$	$17$	$- 30$
	\xa0	$6$		\xa0
	$1$	$- 2$

Multiply the newest entry in the result $(- 2)$ by the divisor $(6)$ and place the result of $(- 12)$ under the next term in the dividend $(17)$ .

$6$	$1$	$- 8$	$17$	$- 30$
	\xa0	$6$	$- 12$	\xa0
	$1$	$- 2$

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

$6$	$1$	$- 8$	$17$	$- 30$
	\xa0	$6$	$- 12$	\xa0
	$1$	$- 2$	$5$

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 $(- 30)$ .

$6$	$1$	$- 8$	$17$	$- 30$
	\xa0	$6$	$- 12$	$30$
	$1$	$- 2$	$5$

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

$6$	$1$	$- 8$	$17$	$- 30$
	\xa0	$6$	$- 12$	$30$
	$1$	$- 2$	$5$	$0$

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

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

Simplify the quotient polynomial.

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

Step 2

The remainder from dividing $\frac{x^{3} - 8 x^{2} + 17 x - 30}{x - 6}$ is $0$ , which means that $x - 6$ is a factor for $x^{3} - 8 x^{2} + 17 x - 30$ .

$x - 6$ is a factor for $x^{3} - 8 x^{2} + 17 x - 30$

Step 3

Find all the possible roots for $x^{2} - 2 x + 5$ .

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.

$p = \pm 1, \pm 5$

$q = \pm 1$

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

$\pm 1, \pm 5$

Step 4

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

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

Step 5

The factored polynomial is $(x - 6) (x^{2} - 2 x + 5)$ .

$(x - 6) (x^{2} - 2 x + 5)$

Find the Factors Using the Factor Theorem x^3-8x^2+17x-30 ; x-6

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