Find the Roots/Zeros Using the Rational Roots Test x^3+8x^2-84x-91

$x^{3} + 8 x^{2} - 84 x - 91$

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.

$p = \pm 1, \pm 7, \pm 13, \pm 91$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 7, \pm 13, \pm 91$

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.

${(- 1)}^{3} + 8 {(- 1)}^{2} - 84 \cdot - 1 - 91$

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 $3$ .

$- 1 + 8 {(- 1)}^{2} - 84 \cdot - 1 - 91$

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

$- 1 + 8 \cdot 1 - 84 \cdot - 1 - 91$

Multiply $8$ by $1$ .

$- 1 + 8 - 84 \cdot - 1 - 91$

Multiply $- 84$ by $- 1$ .

$- 1 + 8 + 84 - 91$

Simplify by adding and subtracting.

Add $- 1$ and $8$ .

$7 + 84 - 91$

Add $7$ and $84$ .

$91 - 91$

Subtract $91$ from $91$ .

$0$

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.

$\frac{x^{3} + 8 x^{2} - 84 x - 91}{x + 1}$

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.

$- 1$	$1$	$8$	$- 84$	$- 91$
	\xa0			\xa0

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

$- 1$	$1$	$8$	$- 84$	$- 91$
	\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 $(8)$ .

$- 1$	$1$	$8$	$- 84$	$- 91$
	\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.

$- 1$	$1$	$8$	$- 84$	$- 91$
	\xa0	$- 1$		\xa0
	$1$	$7$

Multiply the newest entry in the result $(7)$ by the divisor $(- 1)$ and place the result of $(- 7)$ under the next term in the dividend $(- 84)$ .

$- 1$	$1$	$8$	$- 84$	$- 91$
	\xa0	$- 1$	$- 7$	\xa0
	$1$	$7$

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

$- 1$	$1$	$8$	$- 84$	$- 91$
	\xa0	$- 1$	$- 7$	\xa0
	$1$	$7$	$- 91$

Multiply the newest entry in the result $(- 91)$ by the divisor $(- 1)$ and place the result of $(91)$ under the next term in the dividend $(- 91)$ .

$- 1$	$1$	$8$	$- 84$	$- 91$
	\xa0	$- 1$	$- 7$	$91$
	$1$	$7$	$- 91$

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

$- 1$	$1$	$8$	$- 84$	$- 91$
	\xa0	$- 1$	$- 7$	$91$
	$1$	$7$	$- 91$	$0$

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

$1 x^{2} + 7 x - 91$

Simplify the quotient polynomial.

$x^{2} + 7 x - 91$

Step 7

Solve the equation to find any remaining roots.

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 7$ , and $c = - 91$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (1 \cdot - 91)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot - 91}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 91$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 91}}{2 \cdot 1}$

Multiply $- 4$ by $- 91$ .

$x = \frac{- 7 \pm \sqrt{49 + 364}}{2 \cdot 1}$

Add $49$ and $364$ .

$x = \frac{- 7 \pm \sqrt{413}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{413}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{413}}{2}, - \frac{7 + \sqrt{413}}{2}$

Step 8

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

$(x + 1) (x + \frac{7 - \sqrt{413}}{2}) (x + \frac{7 + \sqrt{413}}{2})$

Step 9

These are the roots (zeros) of the polynomial $x^{3} + 8 x^{2} - 84 x - 91$ .

$x = - 1, - \frac{7 - \sqrt{413}}{2}, - \frac{7 + \sqrt{413}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 1, - \frac{7 - \sqrt{413}}{2}, - \frac{7 + \sqrt{413}}{2}$

Decimal Form:

$x = - 1, 6.66120071 \dots, - 13.66120071 \dots$

Step 11

Find the Roots/Zeros Using the Rational Roots Test x^3+8x^2-84x-91

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