Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3+11x^2-2x-22

$f (x) = x^{3} + 11 x^{2} - 2 x - 22$

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.

$p = \pm 1, \pm 2, \pm 11, \pm 22$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 11, \pm 22$

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.

${(- 11)}^{3} + 11 {(- 11)}^{2} - 2 \cdot - 11 - 22$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 11$ is a root of the polynomial.

Simplify each term.

Raise $- 11$ to the power of $3$ .

$- 1331 + 11 {(- 11)}^{2} - 2 \cdot - 11 - 22$

Raise $- 11$ to the power of $2$ .

$- 1331 + 11 \cdot 121 - 2 \cdot - 11 - 22$

Multiply $11$ by $121$ .

$- 1331 + 1331 - 2 \cdot - 11 - 22$

Multiply $- 2$ by $- 11$ .

$- 1331 + 1331 + 22 - 22$

Simplify by adding and subtracting.

Add $- 1331$ and $1331$ .

$0 + 22 - 22$

Add $0$ and $22$ .

$22 - 22$

Subtract $22$ from $22$ .

$0$

Step 5

Since $- 11$ is a known root, divide the polynomial by $x + 11$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 11 x^{2} - 2 x - 22}{x + 11}$

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.

$- 11$	$1$	$11$	$- 2$	$- 22$
	\xa0			\xa0

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

$- 11$	$1$	$11$	$- 2$	$- 22$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 11)$ and place the result of $(- 11)$ under the next term in the dividend $(11)$ .

$- 11$	$1$	$11$	$- 2$	$- 22$
	\xa0	$- 11$		\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.

$- 11$	$1$	$11$	$- 2$	$- 22$
	\xa0	$- 11$		\xa0
	$1$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(- 11)$ and place the result of $(0)$ under the next term in the dividend $(- 2)$ .

$- 11$	$1$	$11$	$- 2$	$- 22$
	\xa0	$- 11$	$0$	\xa0
	$1$	$0$

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

$- 11$	$1$	$11$	$- 2$	$- 22$
	\xa0	$- 11$	$0$	\xa0
	$1$	$0$	$- 2$

Multiply the newest entry in the result $(- 2)$ by the divisor $(- 11)$ and place the result of $(22)$ under the next term in the dividend $(- 22)$ .

$- 11$	$1$	$11$	$- 2$	$- 22$
	\xa0	$- 11$	$0$	$22$
	$1$	$0$	$- 2$

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

$- 11$	$1$	$11$	$- 2$	$- 22$
	\xa0	$- 11$	$0$	$22$
	$1$	$0$	$- 2$	$0$

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

$1 x^{2} + 0 x - 2$

Simplify the quotient polynomial.

$x^{2} - 2$

Step 7

Solve the equation to find any remaining roots.

Add $2$ to both sides of the equation.

$x^{2} = 2$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

Step 8

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

$(x + 11) (x - \sqrt{2}) (x + \sqrt{2})$

Step 9

These are the roots (zeros) of the polynomial $x^{3} + 11 x^{2} - 2 x - 22$ .

$x = - 11, \sqrt{2}, - \sqrt{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 11, \sqrt{2}, - \sqrt{2}$

Decimal Form:

$x = - 11, 1.41421356 \dots, - 1.41421356 \dots$

Step 11

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3+11x^2-2x-22

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