Find the Roots/Zeros Using the Rational Roots Test N(x)=x^3+3x^2+36x+108

$N (x) = x^{3} + 3 x^{2} + 36 x + 108$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 27, \pm 36, \pm 54, \pm 108$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 27, \pm 36, \pm 54, \pm 108$

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.

${(- 3)}^{3} + 3 {(- 3)}^{2} + 36 (- 3) + 108$

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 $3$ .

$- 27 + 3 {(- 3)}^{2} + 36 (- 3) + 108$

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

$- 27 + 3 \cdot 9 + 36 (- 3) + 108$

Multiply $3$ by $9$ .

$- 27 + 27 + 36 (- 3) + 108$

Multiply $36$ by $- 3$ .

$- 27 + 27 - 108 + 108$

Simplify by adding and subtracting.

Add $- 27$ and $27$ .

$0 - 108 + 108$

Subtract $108$ from $0$ .

$- 108 + 108$

Add $- 108$ and $108$ .

$0$

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.

$\frac{x^{3} + 3 x^{2} + 36 x + 108}{x + 3}$

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.

$- 3$	$1$	$3$	$36$	$108$
	\xa0			\xa0

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

$- 3$	$1$	$3$	$36$	$108$
	\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 $(3)$ .

$- 3$	$1$	$3$	$36$	$108$
	\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.

$- 3$	$1$	$3$	$36$	$108$
	\xa0	$- 3$		\xa0
	$1$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(- 3)$ and place the result of $(0)$ under the next term in the dividend $(36)$ .

$- 3$	$1$	$3$	$36$	$108$
	\xa0	$- 3$	$0$	\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.

$- 3$	$1$	$3$	$36$	$108$
	\xa0	$- 3$	$0$	\xa0
	$1$	$0$	$36$

Multiply the newest entry in the result $(36)$ by the divisor $(- 3)$ and place the result of $(- 108)$ under the next term in the dividend $(108)$ .

$- 3$	$1$	$3$	$36$	$108$
	\xa0	$- 3$	$0$	$- 108$
	$1$	$0$	$36$

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

$- 3$	$1$	$3$	$36$	$108$
	\xa0	$- 3$	$0$	$- 108$
	$1$	$0$	$36$	$0$

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

$1 x^{2} + 0 x + 36$

Simplify the quotient polynomial.

$x^{2} + 36$

Step 7

Solve the equation to find any remaining roots.

Subtract $36$ from both sides of the equation.

$x^{2} = - 36$

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

$x = \pm \sqrt{- 36}$

Simplify $\pm \sqrt{- 36}$ .

Rewrite $- 36$ as $- 1 (36)$ .

$x = \pm \sqrt{- 1 (36)}$

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{36}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{36}$

Rewrite $36$ as $6^{2}$ .

$x = \pm i \cdot \sqrt{6^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 6$

Move $6$ to the left of $i$ .

$x = \pm 6 i$

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.

$x = 6 i$

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

$x = - 6 i$

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

$x = 6 i, - 6 i$

Step 8

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

$(x + 3) (x - 6 i) (x + 6 i)$

Step 9

These are the roots (zeros) of the polynomial $x^{3} + 3 x^{2} + 36 x + 108$ .

$x = - 3, 6 i, - 6 i$

Step 10

Find the Roots/Zeros Using the Rational Roots Test N(x)=x^3+3x^2+36x+108

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