Find the Roots/Zeros Using the Rational Roots Test x^4-7x^3-32x^2-32x-8

$x^{4} - 7 x^{3} - 32 x^{2} - 32 x - 8$

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.

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

$q = \pm 1$

Step 2

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

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

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.

${(- 1)}^{4} - 7 {(- 1)}^{3} - 32 {(- 1)}^{2} - 32 \cdot - 1 - 8$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 1$ is a root of the polynomial.

Simplify each term.

Raise $- 1$ to the power of $4$ .

$1 - 7 {(- 1)}^{3} - 32 {(- 1)}^{2} - 32 \cdot - 1 - 8$

Raise $- 1$ to the power of $3$ .

$1 - 7 \cdot - 1 - 32 {(- 1)}^{2} - 32 \cdot - 1 - 8$

Multiply $- 7$ by $- 1$ .

$1 + 7 - 32 {(- 1)}^{2} - 32 \cdot - 1 - 8$

Raise $- 1$ to the power of $2$ .

$1 + 7 - 32 \cdot 1 - 32 \cdot - 1 - 8$

Multiply $- 32$ by $1$ .

$1 + 7 - 32 - 32 \cdot - 1 - 8$

Multiply $- 32$ by $- 1$ .

$1 + 7 - 32 + 32 - 8$

Simplify by adding and subtracting.

Add $1$ and $7$ .

$8 - 32 + 32 - 8$

Subtract $32$ from $8$ .

$- 24 + 32 - 8$

Add $- 24$ and $32$ .

$8 - 8$

Subtract $8$ from $8$ .

$0$

Step 5

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 7 x^{3} - 32 x^{2} - 32 x - 8}{x + 1}$

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.

$- 1$	$1$	$- 7$	$- 32$	$- 32$	$- 8$
	\xa0				\xa0

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

$- 1$	$1$	$- 7$	$- 32$	$- 32$	$- 8$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 1)$ and place the result of $(- 1)$ under the next term in the dividend $(- 7)$ .

$- 1$	$1$	$- 7$	$- 32$	$- 32$	$- 8$
	\xa0	$- 1$			\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.

$- 1$	$1$	$- 7$	$- 32$	$- 32$	$- 8$
	\xa0	$- 1$			\xa0
	$1$	$- 8$

Multiply the newest entry in the result $(- 8)$ by the divisor $(- 1)$ and place the result of $(8)$ under the next term in the dividend $(- 32)$ .

$- 1$	$1$	$- 7$	$- 32$	$- 32$	$- 8$
	\xa0	$- 1$	$8$		\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.

$- 1$	$1$	$- 7$	$- 32$	$- 32$	$- 8$
	\xa0	$- 1$	$8$		\xa0
	$1$	$- 8$	$- 24$

Multiply the newest entry in the result $(- 24)$ by the divisor $(- 1)$ and place the result of $(24)$ under the next term in the dividend $(- 32)$ .

$- 1$	$1$	$- 7$	$- 32$	$- 32$	$- 8$
	\xa0	$- 1$	$8$	$24$	\xa0
	$1$	$- 8$	$- 24$

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

$- 1$	$1$	$- 7$	$- 32$	$- 32$	$- 8$
	\xa0	$- 1$	$8$	$24$	\xa0
	$1$	$- 8$	$- 24$	$- 8$

Multiply the newest entry in the result $(- 8)$ by the divisor $(- 1)$ and place the result of $(8)$ under the next term in the dividend $(- 8)$ .

$- 1$	$1$	$- 7$	$- 32$	$- 32$	$- 8$
	\xa0	$- 1$	$8$	$24$	$8$
	$1$	$- 8$	$- 24$	$- 8$

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

$- 1$	$1$	$- 7$	$- 32$	$- 32$	$- 8$
	\xa0	$- 1$	$8$	$24$	$8$
	$1$	$- 8$	$- 24$	$- 8$	$0$

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

$1 x^{3} + - 8 x^{2} + (- 24) x - 8$

Simplify the quotient polynomial.

$x^{3} - 8 x^{2} - 24 x - 8$

Step 7

Solve the equation to find any remaining roots.

Factor $x^{3} - 8 x^{2} - 24 x - 8$ 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.

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

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

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

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} - 8 {(- 2)}^{2} - 24 \cdot - 2 - 8$

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

$- 8 - 8 {(- 2)}^{2} - 24 \cdot - 2 - 8$

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

$- 8 - 8 \cdot 4 - 24 \cdot - 2 - 8$

Multiply $- 8$ by $4$ .

$- 8 - 32 - 24 \cdot - 2 - 8$

Subtract $32$ from $- 8$ .

$- 40 - 24 \cdot - 2 - 8$

Multiply $- 24$ by $- 2$ .

$- 40 + 48 - 8$

Add $- 40$ and $48$ .

$8 - 8$

Subtract $8$ from $8$ .

$0$

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.

$\frac{x^{3} - 8 x^{2} - 24 x - 8}{x + 2}$

Divide $x^{3} - 8 x^{2} - 24 x - 8$ by $x + 2$ .

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

$x$

$2$

$x^{3}$

$8 x^{2}$

$24 x$

$8$

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			+	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	-	$24 x$

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	-	$24 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	-	$24 x$
					-	$10 x^{2}$	-	$20 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} - 20 x$

				$x^{2}$	-	$10 x$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	-	$24 x$
					+	$10 x^{2}$	+	$20 x$

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

				$x^{2}$	-	$10 x$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	-	$24 x$
					+	$10 x^{2}$	+	$20 x$
							-	$4 x$

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

				$x^{2}$	-	$10 x$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	-	$24 x$
					+	$10 x^{2}$	+	$20 x$
							-	$4 x$	-	$8$

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$	-	$4$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	-	$24 x$
					+	$10 x^{2}$	+	$20 x$
							-	$4 x$	-	$8$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$	-	$4$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	-	$24 x$
					+	$10 x^{2}$	+	$20 x$
							-	$4 x$	-	$8$
							-	$4 x$	-	$8$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x - 8$

				$x^{2}$	-	$10 x$	-	$4$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	-	$24 x$
					+	$10 x^{2}$	+	$20 x$
							-	$4 x$	-	$8$
							+	$4 x$	+	$8$

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

				$x^{2}$	-	$10 x$	-	$4$
$x$	+	$2$		$x^{3}$	-	$8 x^{2}$	-	$24 x$	-	$8$
			-	$x^{3}$	-	$2 x^{2}$
					-	$10 x^{2}$	-	$24 x$
					+	$10 x^{2}$	+	$20 x$
							-	$4 x$	-	$8$
							+	$4 x$	+	$8$
										$0$

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

$x^{2} - 10 x - 4$

Write $x^{3} - 8 x^{2} - 24 x - 8$ as a set of factors.

$(x + 2) (x^{2} - 10 x - 4) = 0$

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

$x + 2 = 0$

$x^{2} - 10 x - 4 = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x^{2} - 10 x - 4$ equal to $0$ and solve for $x$ .

Set $x^{2} - 10 x - 4$ equal to $0$ .

$x^{2} - 10 x - 4 = 0$

Solve $x^{2} - 10 x - 4 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 10$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot - 4)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 4}}{2 \cdot 1}$

Multiply $- 4$ by $- 4$ .

$x = \frac{10 \pm \sqrt{100 + 16}}{2 \cdot 1}$

Add $100$ and $16$ .

$x = \frac{10 \pm \sqrt{116}}{2 \cdot 1}$

Rewrite $116$ as $2^{2} \cdot 29$ .

Factor $4$ out of $116$ .

$x = \frac{10 \pm \sqrt{4 (29)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{10 \pm \sqrt{2^{2} \cdot 29}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 2 \sqrt{29}}{2}$

Simplify $\frac{10 \pm 2 \sqrt{29}}{2}$ .

$x = 5 \pm \sqrt{29}$

The final answer is the combination of both solutions.

$x = 5 + \sqrt{29}, 5 - \sqrt{29}$

The final solution is all the values that make $(x + 2) (x^{2} - 10 x - 4) = 0$ true.

$x = - 2, 5 + \sqrt{29}, 5 - \sqrt{29}$

Step 8

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

$(x + 1) (x + 2) (x - 5 - \sqrt{29}) (x - 5 + \sqrt{29})$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 7 x^{3} - 32 x^{2} - 32 x - 8$ .

$x = - 1, - 2, 5 + \sqrt{29}, 5 - \sqrt{29}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 1, - 2, 5 + \sqrt{29}, 5 - \sqrt{29}$

Decimal Form:

$x = - 1, - 2, 10.38516480 \dots, - 0.38516480 \dots$

Step 11

Find the Roots/Zeros Using the Rational Roots Test x^4-7x^3-32x^2-32x-8

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