Find the Roots/Zeros Using the Rational Roots Test x^4-8x^3-108x^2-241x-130

$x^{4} - 8 x^{3} - 108 x^{2} - 241 x - 130$

Step 1

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.

$p = \pm 1, \pm 2, \pm 5, \pm 10, \pm 13, \pm 26, \pm 65, \pm 130$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 5, \pm 10, \pm 13, \pm 26, \pm 65, \pm 130$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(- 2)}^{4} - 8 {(- 2)}^{3} - 108 {(- 2)}^{2} - 241 \cdot - 2 - 130$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 2$ is a root of the polynomial.

Simplify each term.

Raise $- 2$ to the power of $4$ .

$16 - 8 {(- 2)}^{3} - 108 {(- 2)}^{2} - 241 \cdot - 2 - 130$

Raise $- 2$ to the power of $3$ .

$16 - 8 \cdot - 8 - 108 {(- 2)}^{2} - 241 \cdot - 2 - 130$

Multiply $- 8$ by $- 8$ .

$16 + 64 - 108 {(- 2)}^{2} - 241 \cdot - 2 - 130$

Raise $- 2$ to the power of $2$ .

$16 + 64 - 108 \cdot 4 - 241 \cdot - 2 - 130$

Multiply $- 108$ by $4$ .

$16 + 64 - 432 - 241 \cdot - 2 - 130$

Multiply $- 241$ by $- 2$ .

$16 + 64 - 432 + 482 - 130$

Simplify by adding and subtracting.

Add $16$ and $64$ .

$80 - 432 + 482 - 130$

Subtract $432$ from $80$ .

$- 352 + 482 - 130$

Add $- 352$ and $482$ .

$130 - 130$

Subtract $130$ from $130$ .

$0$

Step 5

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 8 x^{3} - 108 x^{2} - 241 x - 130}{x + 2}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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

$- 2$	$1$	$- 8$	$- 108$	$- 241$	$- 130$
	\xa0				\xa0

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

$- 2$	$1$	$- 8$	$- 108$	$- 241$	$- 130$
	\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 $(- 8)$ .

$- 2$	$1$	$- 8$	$- 108$	$- 241$	$- 130$
	\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.

$- 2$	$1$	$- 8$	$- 108$	$- 241$	$- 130$
	\xa0	$- 2$			\xa0
	$1$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(- 2)$ and place the result of $(20)$ under the next term in the dividend $(- 108)$ .

$- 2$	$1$	$- 8$	$- 108$	$- 241$	$- 130$
	\xa0	$- 2$	$20$		\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.

$- 2$	$1$	$- 8$	$- 108$	$- 241$	$- 130$
	\xa0	$- 2$	$20$		\xa0
	$1$	$- 10$	$- 88$

Multiply the newest entry in the result $(- 88)$ by the divisor $(- 2)$ and place the result of $(176)$ under the next term in the dividend $(- 241)$ .

$- 2$	$1$	$- 8$	$- 108$	$- 241$	$- 130$
	\xa0	$- 2$	$20$	$176$	\xa0
	$1$	$- 10$	$- 88$

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

$- 2$	$1$	$- 8$	$- 108$	$- 241$	$- 130$
	\xa0	$- 2$	$20$	$176$	\xa0
	$1$	$- 10$	$- 88$	$- 65$

Multiply the newest entry in the result $(- 65)$ by the divisor $(- 2)$ and place the result of $(130)$ under the next term in the dividend $(- 130)$ .

$- 2$	$1$	$- 8$	$- 108$	$- 241$	$- 130$
	\xa0	$- 2$	$20$	$176$	$130$
	$1$	$- 10$	$- 88$	$- 65$

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

$- 2$	$1$	$- 8$	$- 108$	$- 241$	$- 130$
	\xa0	$- 2$	$20$	$176$	$130$
	$1$	$- 10$	$- 88$	$- 65$	$0$

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

$1 x^{3} + - 10 x^{2} + (- 88) x - 65$

Simplify the quotient polynomial.

$x^{3} - 10 x^{2} - 88 x - 65$

Step 7

Solve the equation to find any remaining roots.

Factor $x^{3} - 10 x^{2} - 88 x - 65$ using the rational roots test.

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.

$p = \pm 1, \pm 65, \pm 5, \pm 13 q = \pm 1$

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

$\pm 1, \pm 65, \pm 5, \pm 13$

Substitute $- 5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 5$ is a root of the polynomial.

Substitute $- 5$ into the polynomial.

${(- 5)}^{3} - 10 {(- 5)}^{2} - 88 \cdot - 5 - 65$

Raise $- 5$ to the power of $3$ .

$- 125 - 10 {(- 5)}^{2} - 88 \cdot - 5 - 65$

Raise $- 5$ to the power of $2$ .

$- 125 - 10 \cdot 25 - 88 \cdot - 5 - 65$

Multiply $- 10$ by $25$ .

$- 125 - 250 - 88 \cdot - 5 - 65$

Subtract $250$ from $- 125$ .

$- 375 - 88 \cdot - 5 - 65$

Multiply $- 88$ by $- 5$ .

$- 375 + 440 - 65$

Add $- 375$ and $440$ .

$65 - 65$

Subtract $65$ from $65$ .

$0$

Since $- 5$ is a known root, divide the polynomial by $x + 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 10 x^{2} - 88 x - 65}{x + 5}$

Divide $x^{3} - 10 x^{2} - 88 x - 65$ by $x + 5$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$5$

$x^{3}$

$10 x^{2}$

$88 x$

$65$

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			+	$x^{3}$	+	$5 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 5 x^{2}$

				$x^{2}$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$
					-	$15 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$
					-	$15 x^{2}$	-	$88 x$

Divide the highest order term in the dividend $- 15 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$15 x$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$
					-	$15 x^{2}$	-	$88 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$15 x$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$
					-	$15 x^{2}$	-	$88 x$
					-	$15 x^{2}$	-	$75 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{2} - 75 x$

				$x^{2}$	-	$15 x$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$
					-	$15 x^{2}$	-	$88 x$
					+	$15 x^{2}$	+	$75 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$15 x$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$
					-	$15 x^{2}$	-	$88 x$
					+	$15 x^{2}$	+	$75 x$
							-	$13 x$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$15 x$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$
					-	$15 x^{2}$	-	$88 x$
					+	$15 x^{2}$	+	$75 x$
							-	$13 x$	-	$65$

Divide the highest order term in the dividend $- 13 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$15 x$	-	$13$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$
					-	$15 x^{2}$	-	$88 x$
					+	$15 x^{2}$	+	$75 x$
							-	$13 x$	-	$65$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$15 x$	-	$13$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$
					-	$15 x^{2}$	-	$88 x$
					+	$15 x^{2}$	+	$75 x$
							-	$13 x$	-	$65$
							-	$13 x$	-	$65$

The expression needs to be subtracted from the dividend, so change all the signs in $- 13 x - 65$

				$x^{2}$	-	$15 x$	-	$13$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$
					-	$15 x^{2}$	-	$88 x$
					+	$15 x^{2}$	+	$75 x$
							-	$13 x$	-	$65$
							+	$13 x$	+	$65$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$15 x$	-	$13$
$x$	+	$5$		$x^{3}$	-	$10 x^{2}$	-	$88 x$	-	$65$
			-	$x^{3}$	-	$5 x^{2}$
					-	$15 x^{2}$	-	$88 x$
					+	$15 x^{2}$	+	$75 x$
							-	$13 x$	-	$65$
							+	$13 x$	+	$65$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 15 x - 13$

Write $x^{3} - 10 x^{2} - 88 x - 65$ as a set of factors.

$(x + 5) (x^{2} - 15 x - 13) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 5 = 0$

$x^{2} - 15 x - 13 = 0$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Set $x^{2} - 15 x - 13$ equal to $0$ and solve for $x$ .

Set $x^{2} - 15 x - 13$ equal to $0$ .

$x^{2} - 15 x - 13 = 0$

Solve $x^{2} - 15 x - 13 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 15$ , and $c = - 13$ into the quadratic formula and solve for $x$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (1 \cdot - 13)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \cdot - 13}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 13$ .

Multiply $- 4$ by $1$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot - 13}}{2 \cdot 1}$

Multiply $- 4$ by $- 13$ .

$x = \frac{15 \pm \sqrt{225 + 52}}{2 \cdot 1}$

Add $225$ and $52$ .

$x = \frac{15 \pm \sqrt{277}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{15 \pm \sqrt{277}}{2}$

The final answer is the combination of both solutions.

$x = \frac{15 + \sqrt{277}}{2}, \frac{15 - \sqrt{277}}{2}$

The final solution is all the values that make $(x + 5) (x^{2} - 15 x - 13) = 0$ true.

$x = - 5, \frac{15 + \sqrt{277}}{2}, \frac{15 - \sqrt{277}}{2}$

Step 8

The polynomial can be written as a set of linear factors.

$(x + 2) (x + 5) (x - \frac{15 + \sqrt{277}}{2}) (x - \frac{15 - \sqrt{277}}{2})$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 8 x^{3} - 108 x^{2} - 241 x - 130$ .

$x = - 2, - 5, \frac{15 + \sqrt{277}}{2}, \frac{15 - \sqrt{277}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 2, - 5, \frac{15 + \sqrt{277}}{2}, \frac{15 - \sqrt{277}}{2}$

Decimal Form:

$x = - 2, - 5, 15.82165848 \dots, - 0.82165848 \dots$

Step 11

Find the Roots/Zeros Using the Rational Roots Test x^4-8x^3-108x^2-241x-130

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