Find the Factors Using the Factor Theorem x-5 , x^3+5x^2-34x-80

$x - 5$ , $x^{3} + 5 x^{2} - 34 x - 80$

Step 1

Divide $\frac{x^{3} + 5 x^{2} - 34 x - 80}{x - 5}$ 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 $x^{3} + 5 x^{2} - 34 x - 80$ . If the remainder is not equal to $0$ , it means that $x - 5$ is not a factor for $x^{3} + 5 x^{2} - 34 x - 80$ .

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

$5$	$1$	$5$	$- 34$	$- 80$
	\xa0			\xa0

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

$5$	$1$	$5$	$- 34$	$- 80$
	\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 $(5)$ .

$5$	$1$	$5$	$- 34$	$- 80$
	\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.

$5$	$1$	$5$	$- 34$	$- 80$
	\xa0	$5$		\xa0
	$1$	$10$

Multiply the newest entry in the result $(10)$ by the divisor $(5)$ and place the result of $(50)$ under the next term in the dividend $(- 34)$ .

$5$	$1$	$5$	$- 34$	$- 80$
	\xa0	$5$	$50$	\xa0
	$1$	$10$

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

$5$	$1$	$5$	$- 34$	$- 80$
	\xa0	$5$	$50$	\xa0
	$1$	$10$	$16$

Multiply the newest entry in the result $(16)$ by the divisor $(5)$ and place the result of $(80)$ under the next term in the dividend $(- 80)$ .

$5$	$1$	$5$	$- 34$	$- 80$
	\xa0	$5$	$50$	$80$
	$1$	$10$	$16$

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

$5$	$1$	$5$	$- 34$	$- 80$
	\xa0	$5$	$50$	$80$
	$1$	$10$	$16$	$0$

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

$1 x^{2} + 10 x + 16$

Simplify the quotient polynomial.

$x^{2} + 10 x + 16$

Step 2

The remainder from dividing $\frac{x^{3} + 5 x^{2} - 34 x - 80}{x - 5}$ is $0$ , which means that $x - 5$ is a factor for $x^{3} + 5 x^{2} - 34 x - 80$ .

$x - 5$ is a factor for $x^{3} + 5 x^{2} - 34 x - 80$

Step 3

Find all the possible roots for $x^{2} + 10 x + 16$ .

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.

$p = \pm 1, \pm 2, \pm 4, \pm 8, \pm 16$

$q = \pm 1$

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

$\pm 1, \pm 2, \pm 4, \pm 8, \pm 16$

Step 4

Set up the next division to determine if $x + 2$ is a factor of the polynomial $x^{2} + 10 x + 16$ .

$\frac{x^{2} + 10 x + 16}{x + 2}$

Step 5

Divide the expression using synthetic division to determine if it is a factor of the polynomial. Since $x + 2$ divides evenly into $x^{2} + 10 x + 16$ , $x + 2$ is a factor of the polynomial and there is a remaining polynomial of $x + 8$ .

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

$- 2$	$1$	$10$	$16$
	\xa0		\xa0

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

$- 2$	$1$	$10$	$16$
	\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 $(10)$ .

$- 2$	$1$	$10$	$16$
	\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.

$- 2$	$1$	$10$	$16$
	\xa0	$- 2$	\xa0
	$1$	$8$

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 $(16)$ .

$- 2$	$1$	$10$	$16$
	\xa0	$- 2$	$- 16$
	$1$	$8$

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

$- 2$	$1$	$10$	$16$
	\xa0	$- 2$	$- 16$
	$1$	$8$	$0$

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

$(1) x + 8$

Simplify the quotient polynomial.

$x + 8$

Step 6

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

$x + 8$

Step 7

The factored polynomial is $(x - 5) (x + 2) (x + 8)$ .

$(x - 5) (x + 2) (x + 8)$

Find the Factors Using the Factor Theorem x-5 , x^3+5x^2-34x-80

Related Questions

Support LCMGCF.com by disabling your adblocker.