Find the Factors Using the Factor Theorem x+2 , x^3+4x^2-4x-16

$x + 2$ , $x^{3} + 4 x^{2} - 4 x - 16$

Step 1

Divide $\frac{x^{3} + 4 x^{2} - 4 x - 16}{x + 2}$ 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 $x^{3} + 4 x^{2} - 4 x - 16$ . If the remainder is not equal to $0$ , it means that $x + 2$ is not a factor for $x^{3} + 4 x^{2} - 4 x - 16$ .

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

$- 2$	$1$	$4$	$- 4$	$- 16$
	\xa0			\xa0

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

$- 2$	$1$	$4$	$- 4$	$- 16$
	\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 $(4)$ .

$- 2$	$1$	$4$	$- 4$	$- 16$
	\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.

$- 2$	$1$	$4$	$- 4$	$- 16$
	\xa0	$- 2$		\xa0
	$1$	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(- 2)$ and place the result of $(- 4)$ under the next term in the dividend $(- 4)$ .

$- 2$	$1$	$4$	$- 4$	$- 16$
	\xa0	$- 2$	$- 4$	\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.

$- 2$	$1$	$4$	$- 4$	$- 16$
	\xa0	$- 2$	$- 4$	\xa0
	$1$	$2$	$- 8$

Multiply the newest entry in the result $(- 8)$ by the divisor $(- 2)$ and place the result of $(16)$ under the next term in the dividend $(- 16)$ .

$- 2$	$1$	$4$	$- 4$	$- 16$
	\xa0	$- 2$	$- 4$	$16$
	$1$	$2$	$- 8$

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

$- 2$	$1$	$4$	$- 4$	$- 16$
	\xa0	$- 2$	$- 4$	$16$
	$1$	$2$	$- 8$	$0$

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

$1 x^{2} + 2 x - 8$

Simplify the quotient polynomial.

$x^{2} + 2 x - 8$

Step 2

The remainder from dividing $\frac{x^{3} + 4 x^{2} - 4 x - 16}{x + 2}$ is $0$ , which means that $x + 2$ is a factor for $x^{3} + 4 x^{2} - 4 x - 16$ .

$x + 2$ is a factor for $x^{3} + 4 x^{2} - 4 x - 16$

Step 3

Find all the possible roots for $x^{2} + 2 x - 8$ .

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.

$p = \pm 1, \pm 2, \pm 4, \pm 8$

$q = \pm 1$

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

$\pm 1, \pm 2, \pm 4, \pm 8$

Step 4

Set up the next division to determine if $x - 2$ is a factor of the polynomial $x^{2} + 2 x - 8$ .

$\frac{x^{2} + 2 x - 8}{x - 2}$

Step 5

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x - 2$ divides evenly into $x^{2} + 2 x - 8$ , $x - 2$ 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.

$2$	$1$	$2$	$- 8$
	\xa0		\xa0

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

$2$	$1$	$2$	$- 8$
	\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 $(2)$ .

$2$	$1$	$2$	$- 8$
	\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.

$2$	$1$	$2$	$- 8$
	\xa0	$2$	\xa0
	$1$	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(2)$ and place the result of $(8)$ under the next term in the dividend $(- 8)$ .

$2$	$1$	$2$	$- 8$
	\xa0	$2$	$8$
	$1$	$4$

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

$2$	$1$	$2$	$- 8$
	\xa0	$2$	$8$
	$1$	$4$	$0$

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

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

Find the Factors Using the Factor Theorem x+2 , x^3+4x^2-4x-16

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