Find the Factors Using the Factor Theorem x+3 , x^3+5x^2-9x-45

$x + 3$ , $x^{3} + 5 x^{2} - 9 x - 45$

Step 1

Divide $\frac{x^{3} + 5 x^{2} - 9 x - 45}{x + 3}$ using synthetic division and check if the remainder is equal to $0$ . If the remainder is equal to $0$ , it means that $x + 3$ is a factor for $x^{3} + 5 x^{2} - 9 x - 45$ . If the remainder is not equal to $0$ , it means that $x + 3$ is not a factor for $x^{3} + 5 x^{2} - 9 x - 45$ .

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

$- 3$	$1$	$5$	$- 9$	$- 45$
	\xa0			\xa0

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

$- 3$	$1$	$5$	$- 9$	$- 45$
	\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 $(5)$ .

$- 3$	$1$	$5$	$- 9$	$- 45$
	\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.

$- 3$	$1$	$5$	$- 9$	$- 45$
	\xa0	$- 3$		\xa0
	$1$	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(- 3)$ and place the result of $(- 6)$ under the next term in the dividend $(- 9)$ .

$- 3$	$1$	$5$	$- 9$	$- 45$
	\xa0	$- 3$	$- 6$	\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.

$- 3$	$1$	$5$	$- 9$	$- 45$
	\xa0	$- 3$	$- 6$	\xa0
	$1$	$2$	$- 15$

Multiply the newest entry in the result $(- 15)$ by the divisor $(- 3)$ and place the result of $(45)$ under the next term in the dividend $(- 45)$ .

$- 3$	$1$	$5$	$- 9$	$- 45$
	\xa0	$- 3$	$- 6$	$45$
	$1$	$2$	$- 15$

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

$- 3$	$1$	$5$	$- 9$	$- 45$
	\xa0	$- 3$	$- 6$	$45$
	$1$	$2$	$- 15$	$0$

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

$1 x^{2} + 2 x - 15$

Simplify the quotient polynomial.

$x^{2} + 2 x - 15$

Step 2

The remainder from dividing $\frac{x^{3} + 5 x^{2} - 9 x - 45}{x + 3}$ is $0$ , which means that $x + 3$ is a factor for $x^{3} + 5 x^{2} - 9 x - 45$ .

$x + 3$ is a factor for $x^{3} + 5 x^{2} - 9 x - 45$

Step 3

Find all the possible roots for $x^{2} + 2 x - 15$ .

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.

$p = \pm 1, \pm 3, \pm 5, \pm 15$

$q = \pm 1$

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

$\pm 1, \pm 3, \pm 5, \pm 15$

Step 4

Set up the next division to determine if $x - 3$ is a factor of the polynomial $x^{2} + 2 x - 15$ .

$\frac{x^{2} + 2 x - 15}{x - 3}$

Step 5

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x - 3$ divides evenly into $x^{2} + 2 x - 15$ , $x - 3$ 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.

$3$	$1$	$2$	$- 15$
	\xa0		\xa0

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

$3$	$1$	$2$	$- 15$
	\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 $(2)$ .

$3$	$1$	$2$	$- 15$
	\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.

$3$	$1$	$2$	$- 15$
	\xa0	$3$	\xa0
	$1$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(3)$ and place the result of $(15)$ under the next term in the dividend $(- 15)$ .

$3$	$1$	$2$	$- 15$
	\xa0	$3$	$15$
	$1$	$5$

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

$3$	$1$	$2$	$- 15$
	\xa0	$3$	$15$
	$1$	$5$	$0$

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

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

Find the Factors Using the Factor Theorem x+3 , x^3+5x^2-9x-45

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