Find the Roots/Zeros Using the Rational Roots Test x^4-5x^3-100x^2-283x-165

$x^{4} - 5 x^{3} - 100 x^{2} - 283 x - 165$

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.

$p = \pm 1, \pm 3, \pm 5, \pm 11, \pm 15, \pm 33, \pm 55, \pm 165$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 3, \pm 5, \pm 11, \pm 15, \pm 33, \pm 55, \pm 165$

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.

${(- 3)}^{4} - 5 {(- 3)}^{3} - 100 {(- 3)}^{2} - 283 \cdot - 3 - 165$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 3$ is a root of the polynomial.

Simplify each term.

Raise $- 3$ to the power of $4$ .

$81 - 5 {(- 3)}^{3} - 100 {(- 3)}^{2} - 283 \cdot - 3 - 165$

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

$81 - 5 \cdot - 27 - 100 {(- 3)}^{2} - 283 \cdot - 3 - 165$

Multiply $- 5$ by $- 27$ .

$81 + 135 - 100 {(- 3)}^{2} - 283 \cdot - 3 - 165$

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

$81 + 135 - 100 \cdot 9 - 283 \cdot - 3 - 165$

Multiply $- 100$ by $9$ .

$81 + 135 - 900 - 283 \cdot - 3 - 165$

Multiply $- 283$ by $- 3$ .

$81 + 135 - 900 + 849 - 165$

Simplify by adding and subtracting.

Add $81$ and $135$ .

$216 - 900 + 849 - 165$

Subtract $900$ from $216$ .

$- 684 + 849 - 165$

Add $- 684$ and $849$ .

$165 - 165$

Subtract $165$ from $165$ .

$0$

Step 5

Since $- 3$ is a known root, divide the polynomial by $x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 5 x^{3} - 100 x^{2} - 283 x - 165}{x + 3}$

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.

$- 3$	$1$	$- 5$	$- 100$	$- 283$	$- 165$
	\xa0				\xa0

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

$- 3$	$1$	$- 5$	$- 100$	$- 283$	$- 165$
	\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 $(- 5)$ .

$- 3$	$1$	$- 5$	$- 100$	$- 283$	$- 165$
	\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.

$- 3$	$1$	$- 5$	$- 100$	$- 283$	$- 165$
	\xa0	$- 3$			\xa0
	$1$	$- 8$

Multiply the newest entry in the result $(- 8)$ by the divisor $(- 3)$ and place the result of $(24)$ under the next term in the dividend $(- 100)$ .

$- 3$	$1$	$- 5$	$- 100$	$- 283$	$- 165$
	\xa0	$- 3$	$24$		\xa0
	$1$	$- 8$

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

$- 3$	$1$	$- 5$	$- 100$	$- 283$	$- 165$
	\xa0	$- 3$	$24$		\xa0
	$1$	$- 8$	$- 76$

Multiply the newest entry in the result $(- 76)$ by the divisor $(- 3)$ and place the result of $(228)$ under the next term in the dividend $(- 283)$ .

$- 3$	$1$	$- 5$	$- 100$	$- 283$	$- 165$
	\xa0	$- 3$	$24$	$228$	\xa0
	$1$	$- 8$	$- 76$

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

$- 3$	$1$	$- 5$	$- 100$	$- 283$	$- 165$
	\xa0	$- 3$	$24$	$228$	\xa0
	$1$	$- 8$	$- 76$	$- 55$

Multiply the newest entry in the result $(- 55)$ by the divisor $(- 3)$ and place the result of $(165)$ under the next term in the dividend $(- 165)$ .

$- 3$	$1$	$- 5$	$- 100$	$- 283$	$- 165$
	\xa0	$- 3$	$24$	$228$	$165$
	$1$	$- 8$	$- 76$	$- 55$

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

$- 3$	$1$	$- 5$	$- 100$	$- 283$	$- 165$
	\xa0	$- 3$	$24$	$228$	$165$
	$1$	$- 8$	$- 76$	$- 55$	$0$

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

$1 x^{3} + - 8 x^{2} + (- 76) x - 55$

Simplify the quotient polynomial.

$x^{3} - 8 x^{2} - 76 x - 55$

Step 7

Solve the equation to find any remaining roots.

Factor $x^{3} - 8 x^{2} - 76 x - 55$ 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.

$p = \pm 1, \pm 55, \pm 5, \pm 11 q = \pm 1$

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

$\pm 1, \pm 55, \pm 5, \pm 11$

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.

${(- 5)}^{3} - 8 {(- 5)}^{2} - 76 \cdot - 5 - 55$

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

$- 125 - 8 {(- 5)}^{2} - 76 \cdot - 5 - 55$

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

$- 125 - 8 \cdot 25 - 76 \cdot - 5 - 55$

Multiply $- 8$ by $25$ .

$- 125 - 200 - 76 \cdot - 5 - 55$

Subtract $200$ from $- 125$ .

$- 325 - 76 \cdot - 5 - 55$

Multiply $- 76$ by $- 5$ .

$- 325 + 380 - 55$

Add $- 325$ and $380$ .

$55 - 55$

Subtract $55$ from $55$ .

$0$

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.

$\frac{x^{3} - 8 x^{2} - 76 x - 55}{x + 5}$

Divide $x^{3} - 8 x^{2} - 76 x - 55$ 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 $0$ .

$x$

$5$

$x^{3}$

$8 x^{2}$

$76 x$

$55$

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

					$x^{2}$
	$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			+	$x^{3}$	+	$5 x^{2}$

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

				$x^{2}$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$

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

				$x^{2}$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$
					-	$13 x^{2}$

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

				$x^{2}$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$
					-	$13 x^{2}$	-	$76 x$

Divide the highest order term in the dividend $- 13 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$13 x$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$
					-	$13 x^{2}$	-	$76 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$13 x$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$
					-	$13 x^{2}$	-	$76 x$
					-	$13 x^{2}$	-	$65 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 13 x^{2} - 65 x$

				$x^{2}$	-	$13 x$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$
					-	$13 x^{2}$	-	$76 x$
					+	$13 x^{2}$	+	$65 x$

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

				$x^{2}$	-	$13 x$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$
					-	$13 x^{2}$	-	$76 x$
					+	$13 x^{2}$	+	$65 x$
							-	$11 x$

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

				$x^{2}$	-	$13 x$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$
					-	$13 x^{2}$	-	$76 x$
					+	$13 x^{2}$	+	$65 x$
							-	$11 x$	-	$55$

Divide the highest order term in the dividend $- 11 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$13 x$	-	$11$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$
					-	$13 x^{2}$	-	$76 x$
					+	$13 x^{2}$	+	$65 x$
							-	$11 x$	-	$55$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$13 x$	-	$11$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$
					-	$13 x^{2}$	-	$76 x$
					+	$13 x^{2}$	+	$65 x$
							-	$11 x$	-	$55$
							-	$11 x$	-	$55$

The expression needs to be subtracted from the dividend, so change all the signs in $- 11 x - 55$

				$x^{2}$	-	$13 x$	-	$11$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$
					-	$13 x^{2}$	-	$76 x$
					+	$13 x^{2}$	+	$65 x$
							-	$11 x$	-	$55$
							+	$11 x$	+	$55$

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

				$x^{2}$	-	$13 x$	-	$11$
$x$	+	$5$		$x^{3}$	-	$8 x^{2}$	-	$76 x$	-	$55$
			-	$x^{3}$	-	$5 x^{2}$
					-	$13 x^{2}$	-	$76 x$
					+	$13 x^{2}$	+	$65 x$
							-	$11 x$	-	$55$
							+	$11 x$	+	$55$
										$0$

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

$x^{2} - 13 x - 11$

Write $x^{3} - 8 x^{2} - 76 x - 55$ as a set of factors.

$(x + 5) (x^{2} - 13 x - 11) = 0$

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

$x + 5 = 0$

$x^{2} - 13 x - 11 = 0$

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 $x^{2} - 13 x - 11$ equal to $0$ and solve for $x$ .

Set $x^{2} - 13 x - 11$ equal to $0$ .

$x^{2} - 13 x - 11 = 0$

Solve $x^{2} - 13 x - 11 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 13$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{13 \pm \sqrt{{(- 13)}^{2} - 4 \cdot (1 \cdot - 11)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 11$ .

Multiply $- 4$ by $1$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot - 11}}{2 \cdot 1}$

Multiply $- 4$ by $- 11$ .

$x = \frac{13 \pm \sqrt{169 + 44}}{2 \cdot 1}$

Add $169$ and $44$ .

$x = \frac{13 \pm \sqrt{213}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{13 \pm \sqrt{213}}{2}$

The final answer is the combination of both solutions.

$x = \frac{13 + \sqrt{213}}{2}, \frac{13 - \sqrt{213}}{2}$

The final solution is all the values that make $(x + 5) (x^{2} - 13 x - 11) = 0$ true.

$x = - 5, \frac{13 + \sqrt{213}}{2}, \frac{13 - \sqrt{213}}{2}$

Step 8

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

$(x + 3) (x + 5) (x - \frac{13 + \sqrt{213}}{2}) (x - \frac{13 - \sqrt{213}}{2})$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 5 x^{3} - 100 x^{2} - 283 x - 165$ .

$x = - 3, - 5, \frac{13 + \sqrt{213}}{2}, \frac{13 - \sqrt{213}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 3, - 5, \frac{13 + \sqrt{213}}{2}, \frac{13 - \sqrt{213}}{2}$

Decimal Form:

$x = - 3, - 5, 13.79725975 \dots, - 0.79725975 \dots$

Step 11

Find the Roots/Zeros Using the Rational Roots Test x^4-5x^3-100x^2-283x-165

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