Find the Factors Using the Factor Theorem x+2 , x^3+3x^2-10x-24

$x + 2$ , $x^{3} + 3 x^{2} - 10 x - 24$

Step 1

Divide $\frac{x^{3} + 3 x^{2} - 10 x - 24}{x + 2}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x + 2$ is a factor for $x^{3} + 3 x^{2} - 10 x - 24$ . If the remainder is not equal to $0$ , it means that $x + 2$ is not a factor for $x^{3} + 3 x^{2} - 10 x - 24$ .

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

$- 2$	$1$	$3$	$- 10$	$- 24$
	\xa0			\xa0

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

$- 2$	$1$	$3$	$- 10$	$- 24$
	\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 $(3)$ .

$- 2$	$1$	$3$	$- 10$	$- 24$
	\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.

$- 2$	$1$	$3$	$- 10$	$- 24$
	\xa0	$- 2$		\xa0
	$1$	$1$

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 $(- 10)$ .

$- 2$	$1$	$3$	$- 10$	$- 24$
	\xa0	$- 2$	$- 2$	\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.

$- 2$	$1$	$3$	$- 10$	$- 24$
	\xa0	$- 2$	$- 2$	\xa0
	$1$	$1$	$- 12$

Multiply the newest entry in the result $(- 12)$ by the divisor $(- 2)$ and place the result of $(24)$ under the next term in the dividend $(- 24)$ .

$- 2$	$1$	$3$	$- 10$	$- 24$
	\xa0	$- 2$	$- 2$	$24$
	$1$	$1$	$- 12$

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

$- 2$	$1$	$3$	$- 10$	$- 24$
	\xa0	$- 2$	$- 2$	$24$
	$1$	$1$	$- 12$	$0$

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

$1 x^{2} + 1 x - 12$

Simplify the quotient polynomial.

$x^{2} + x - 12$

Step 2

The remainder from dividing $\frac{x^{3} + 3 x^{2} - 10 x - 24}{x + 2}$ is $0$ , which means that $x + 2$ is a factor for $x^{3} + 3 x^{2} - 10 x - 24$ .

$x + 2$ is a factor for $x^{3} + 3 x^{2} - 10 x - 24$

Step 3

Find all the possible roots for $x^{2} + x - 12$ .

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.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

$q = \pm 1$

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

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

Step 4

Set up the next division to determine if $x - 3$ is a factor of the polynomial $x^{2} + x - 12$ .

$\frac{x^{2} + x - 12}{x - 3}$

Step 5

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x - 3$ divides evenly into $x^{2} + x - 12$ , $x - 3$ is a factor of the polynomial and there is a remaining polynomial of $x + 4$ .

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

$3$	$1$	$1$	$- 12$
	\xa0		\xa0

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

$3$	$1$	$1$	$- 12$
	\xa0		\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(3)$ and place the result of $(3)$ under the next term in the dividend $(1)$ .

$3$	$1$	$1$	$- 12$
	\xa0	$3$	\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.

$3$	$1$	$1$	$- 12$
	\xa0	$3$	\xa0
	$1$	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(3)$ and place the result of $(12)$ under the next term in the dividend $(- 12)$ .

$3$	$1$	$1$	$- 12$
	\xa0	$3$	$12$
	$1$	$4$

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

$3$	$1$	$1$	$- 12$
	\xa0	$3$	$12$
	$1$	$4$	$0$

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

$(1) x + 4$

Simplify the quotient polynomial.

$x + 4$

Step 6

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

$x + 4$

Step 7

The factored polynomial is $(x + 2) (x - 3) (x + 4)$ .

$(x + 2) (x - 3) (x + 4)$

Find the Factors Using the Factor Theorem x+2 , x^3+3x^2-10x-24

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