Find the Factors Using the Factor Theorem x+5 , x^3+6x^2-x-30

$x + 5$ , $x^{3} + 6 x^{2} - x - 30$

Step 1

Divide $\frac{x^{3} + 6 x^{2} - x - 30}{x + 5}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x + 5$ is a factor for $x^{3} + 6 x^{2} - x - 30$ . If the remainder is not equal to $0$ , it means that $x + 5$ is not a factor for $x^{3} + 6 x^{2} - x - 30$ .

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

$- 5$	$1$	$6$	$- 1$	$- 30$
	\xa0			\xa0

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

$- 5$	$1$	$6$	$- 1$	$- 30$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 5)$ and place the result of $(- 5)$ under the next term in the dividend $(6)$ .

$- 5$	$1$	$6$	$- 1$	$- 30$
	\xa0	$- 5$		\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.

$- 5$	$1$	$6$	$- 1$	$- 30$
	\xa0	$- 5$		\xa0
	$1$	$1$

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

$- 5$	$1$	$6$	$- 1$	$- 30$
	\xa0	$- 5$	$- 5$	\xa0
	$1$	$1$

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

$- 5$	$1$	$6$	$- 1$	$- 30$
	\xa0	$- 5$	$- 5$	\xa0
	$1$	$1$	$- 6$

Multiply the newest entry in the result $(- 6)$ by the divisor $(- 5)$ and place the result of $(30)$ under the next term in the dividend $(- 30)$ .

$- 5$	$1$	$6$	$- 1$	$- 30$
	\xa0	$- 5$	$- 5$	$30$
	$1$	$1$	$- 6$

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

$- 5$	$1$	$6$	$- 1$	$- 30$
	\xa0	$- 5$	$- 5$	$30$
	$1$	$1$	$- 6$	$0$

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

$1 x^{2} + 1 x - 6$

Simplify the quotient polynomial.

$x^{2} + x - 6$

Step 2

The remainder from dividing $\frac{x^{3} + 6 x^{2} - x - 30}{x + 5}$ is $0$ , which means that $x + 5$ is a factor for $x^{3} + 6 x^{2} - x - 30$ .

$x + 5$ is a factor for $x^{3} + 6 x^{2} - x - 30$

Step 3

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

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.

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

$q = \pm 1$

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

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

Step 4

Set up the next division to determine if $x - 2$ is a factor of the polynomial $x^{2} + x - 6$ .

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

Step 5

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

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

$2$	$1$	$1$	$- 6$
	\xa0		\xa0

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

$2$	$1$	$1$	$- 6$
	\xa0		\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(2)$ and place the result of $(2)$ under the next term in the dividend $(1)$ .

$2$	$1$	$1$	$- 6$
	\xa0	$2$	\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.

$2$	$1$	$1$	$- 6$
	\xa0	$2$	\xa0
	$1$	$3$

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 $(- 6)$ .

$2$	$1$	$1$	$- 6$
	\xa0	$2$	$6$
	$1$	$3$

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

$2$	$1$	$1$	$- 6$
	\xa0	$2$	$6$
	$1$	$3$	$0$

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

$(1) x + 3$

Simplify the quotient polynomial.

$x + 3$

Step 6

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

$x + 3$

Step 7

The factored polynomial is $(x + 5) (x - 2) (x + 3)$ .

$(x + 5) (x - 2) (x + 3)$

Find the Factors Using the Factor Theorem x+5 , x^3+6x^2-x-30

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