Find the Factors Using the Factor Theorem x^3+10x^2+19x-30 , x+6

$x^{3} + 10 x^{2} + 19 x - 30$ , $x + 6$

Step 1

Divide $\frac{x^{3} + 10 x^{2} + 19 x - 30}{x + 6}$ 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 $x^{3} + 10 x^{2} + 19 x - 30$ . If the remainder is not equal to $0$ , it means that $x + 6$ is not a factor for $x^{3} + 10 x^{2} + 19 x - 30$ .

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

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0			\xa0

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

$- 6$	$1$	$10$	$19$	$- 30$
	\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 $(10)$ .

$- 6$	$1$	$10$	$19$	$- 30$
	\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.

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0	$- 6$		\xa0
	$1$	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- 6)$ and place the result of $(- 24)$ under the next term in the dividend $(19)$ .

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0	$- 6$	$- 24$	\xa0
	$1$	$4$

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

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0	$- 6$	$- 24$	\xa0
	$1$	$4$	$- 5$

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 $(- 30)$ .

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0	$- 6$	$- 24$	$30$
	$1$	$4$	$- 5$

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

$- 6$	$1$	$10$	$19$	$- 30$
	\xa0	$- 6$	$- 24$	$30$
	$1$	$4$	$- 5$	$0$

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

$1 x^{2} + 4 x - 5$

Simplify the quotient polynomial.

$x^{2} + 4 x - 5$

Step 2

The remainder from dividing $\frac{x^{3} + 10 x^{2} + 19 x - 30}{x + 6}$ is $0$ , which means that $x + 6$ is a factor for $x^{3} + 10 x^{2} + 19 x - 30$ .

$x + 6$ is a factor for $x^{3} + 10 x^{2} + 19 x - 30$

Step 3

Find all the possible roots for $x^{2} + 4 x - 5$ .

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

Set up the next division to determine if $x - 1$ is a factor of the polynomial $x^{2} + 4 x - 5$ .

$\frac{x^{2} + 4 x - 5}{x - 1}$

Step 5

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x - 1$ divides evenly into $x^{2} + 4 x - 5$ , $x - 1$ is a factor of the polynomial and there is a remaining polynomial of $x + 5$ .

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

$1$	$1$	$4$	$- 5$
	\xa0		\xa0

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

$1$	$1$	$4$	$- 5$
	\xa0		\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(1)$ and place the result of $(1)$ under the next term in the dividend $(4)$ .

$1$	$1$	$4$	$- 5$
	\xa0	$1$	\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.

$1$	$1$	$4$	$- 5$
	\xa0	$1$	\xa0
	$1$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(1)$ and place the result of $(5)$ under the next term in the dividend $(- 5)$ .

$1$	$1$	$4$	$- 5$
	\xa0	$1$	$5$
	$1$	$5$

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

$1$	$1$	$4$	$- 5$
	\xa0	$1$	$5$
	$1$	$5$	$0$

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

$(1) x + 5$

Simplify the quotient polynomial.

$x + 5$

Step 6

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

$x + 5$

Step 7

The factored polynomial is $(x + 6) (x - 1) (x + 5)$ .

$(x + 6) (x - 1) (x + 5)$

Find the Factors Using the Factor Theorem x^3+10x^2+19x-30 , x+6

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