Find the Factors Using the Factor Theorem 2x^3-5x^2-22x-15 ; x-5

$2 x^{3} - 5 x^{2} - 22 x - 15$ ; $x - 5$

Divide $\frac{2 x^{3} - 5 x^{2} - 22 x - 15}{x - 5}$ 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 $2 x^{3} - 5 x^{2} - 22 x - 15$ . If the remainder is not equal to $0$ , it means that $x - 5$ is not a factor for $2 x^{3} - 5 x^{2} - 22 x - 15$ .

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

$5$	$2$	$- 5$	$- 22$	$- 15$
	\xa0			\xa0

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$5$	$2$	$- 5$	$- 22$	$- 15$
	\xa0			\xa0
	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(5)$ and place the result of $(10)$ under the next term in the dividend $(- 5)$ .

$5$	$2$	$- 5$	$- 22$	$- 15$
	\xa0	$10$		\xa0
	$2$

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

$5$	$2$	$- 5$	$- 22$	$- 15$
	\xa0	$10$		\xa0
	$2$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(5)$ and place the result of $(25)$ under the next term in the dividend $(- 22)$ .

$5$	$2$	$- 5$	$- 22$	$- 15$
	\xa0	$10$	$25$	\xa0
	$2$	$5$

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

$5$	$2$	$- 5$	$- 22$	$- 15$
	\xa0	$10$	$25$	\xa0
	$2$	$5$	$3$

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

$5$	$2$	$- 5$	$- 22$	$- 15$
	\xa0	$10$	$25$	$15$
	$2$	$5$	$3$

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

$5$	$2$	$- 5$	$- 22$	$- 15$
	\xa0	$10$	$25$	$15$
	$2$	$5$	$3$	$0$

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

$2 x^{2} + 5 x + 3$

The remainder from dividing $\frac{2 x^{3} - 5 x^{2} - 22 x - 15}{x - 5}$ is $0$ , which means that $x - 5$ is a factor for $2 x^{3} - 5 x^{2} - 22 x - 15$ .

$x - 5$ is a factor for $2 x^{3} - 5 x^{2} - 22 x - 15$

Find all the possible roots for $2 x^{2} + 5 x + 3$ .

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.

$p = \pm 1, \pm 3$

$q = \pm 1, \pm 2$

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

$\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}$

Set up the next division to determine if $x + 1$ is a factor of the polynomial $2 x^{2} + 5 x + 3$ .

$\frac{2 x^{2} + 5 x + 3}{x + 1}$

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x + 1$ divides evenly into $2 x^{2} + 5 x + 3$ , $x + 1$ is a factor of the polynomial and there is a remaining polynomial of $2 x + 3$ .

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

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

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

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

Multiply the newest entry in the result $(2)$ by the divisor $(- 1)$ and place the result of $(- 2)$ under the next term in the dividend $(5)$ .

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

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

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

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

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

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

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

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

$(2) x + 3$

Simplify the quotient polynomial.

$2 x + 3$

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

$2 x + 3$

The factored polynomial is $(x - 5) (x + 1) (2 x + 3)$ .

$(x - 5) (x + 1) (2 x + 3)$

Find the Factors Using the Factor Theorem 2x^3-5x^2-22x-15 ; x-5

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