Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^3-6x^2-15x+30

$f (x) = 3 x^{3} - 6 x^{2} - 15 x + 30$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30$

$q = \pm 1, \pm 3$

Step 2

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

$\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm 5, \pm \frac{5}{3}, \pm 6, \pm 10, \pm \frac{10}{3}, \pm 15, \pm 30$

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.

$3 {(2)}^{3} - 6 {(2)}^{2} - 15 \cdot 2 + 30$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 2$ is a root of the polynomial.

Simplify each term.

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

$3 \cdot 8 - 6 {(2)}^{2} - 15 \cdot 2 + 30$

Multiply $3$ by $8$ .

$24 - 6 {(2)}^{2} - 15 \cdot 2 + 30$

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

$24 - 6 \cdot 4 - 15 \cdot 2 + 30$

Multiply $- 6$ by $4$ .

$24 - 24 - 15 \cdot 2 + 30$

Multiply $- 15$ by $2$ .

$24 - 24 - 30 + 30$

Simplify by adding and subtracting.

Subtract $24$ from $24$ .

$0 - 30 + 30$

Subtract $30$ from $0$ .

$- 30 + 30$

Add $- 30$ and $30$ .

$0$

Step 5

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.

$\frac{3 x^{3} - 6 x^{2} - 15 x + 30}{x - 2}$

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.

$2$	$3$	$- 6$	$- 15$	$30$
	\xa0			\xa0

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$2$	$3$	$- 6$	$- 15$	$30$
	\xa0			\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(2)$ and place the result of $(6)$ under the next term in the dividend $(- 6)$ .

$2$	$3$	$- 6$	$- 15$	$30$
	\xa0	$6$		\xa0
	$3$

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

$2$	$3$	$- 6$	$- 15$	$30$
	\xa0	$6$		\xa0
	$3$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(2)$ and place the result of $(0)$ under the next term in the dividend $(- 15)$ .

$2$	$3$	$- 6$	$- 15$	$30$
	\xa0	$6$	$0$	\xa0
	$3$	$0$

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

$2$	$3$	$- 6$	$- 15$	$30$
	\xa0	$6$	$0$	\xa0
	$3$	$0$	$- 15$

Multiply the newest entry in the result $(- 15)$ by the divisor $(2)$ and place the result of $(- 30)$ under the next term in the dividend $(30)$ .

$2$	$3$	$- 6$	$- 15$	$30$
	\xa0	$6$	$0$	$- 30$
	$3$	$0$	$- 15$

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

$2$	$3$	$- 6$	$- 15$	$30$
	\xa0	$6$	$0$	$- 30$
	$3$	$0$	$- 15$	$0$

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

$3 x^{2} + 0 x - 15$

Simplify the quotient polynomial.

$3 x^{2} - 15$

Step 7

Factor $3$ out of $3 x^{2} - 15$ .

Factor $3$ out of $3 x^{2}$ .

$(x - 2) (3 (x^{2}) - 15)$

Factor $3$ out of $- 15$ .

$(x - 2) (3 x^{2} + 3 \cdot - 5)$

Factor $3$ out of $3 x^{2} + 3 \cdot - 5$ .

$(x - 2) (3 (x^{2} - 5))$

Step 8

Factor the left side of the equation.

Factor $3$ out of $3 x^{3} - 6 x^{2} - 15 x + 30$ .

Factor $3$ out of $3 x^{3}$ .

$3 (x^{3}) - 6 x^{2} - 15 x + 30 = 0$

Factor $3$ out of $- 6 x^{2}$ .

$3 (x^{3}) + 3 (- 2 x^{2}) - 15 x + 30 = 0$

Factor $3$ out of $- 15 x$ .

$3 (x^{3}) + 3 (- 2 x^{2}) + 3 (- 5 x) + 30 = 0$

Factor $3$ out of $30$ .

$3 x^{3} + 3 (- 2 x^{2}) + 3 (- 5 x) + 3 \cdot 10 = 0$

Factor $3$ out of $3 x^{3} + 3 (- 2 x^{2})$ .

$3 (x^{3} - 2 x^{2}) + 3 (- 5 x) + 3 \cdot 10 = 0$

Factor $3$ out of $3 (x^{3} - 2 x^{2}) + 3 (- 5 x)$ .

$3 (x^{3} - 2 x^{2} - 5 x) + 3 \cdot 10 = 0$

Factor $3$ out of $3 (x^{3} - 2 x^{2} - 5 x) + 3 \cdot 10$ .

$3 (x^{3} - 2 x^{2} - 5 x + 10) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$3 ((x^{3} - 2 x^{2}) - 5 x + 10) = 0$

Factor out the greatest common factor (GCF) from each group.

$3 (x^{2} (x - 2) - 5 (x - 2)) = 0$

Factor.

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$3 ((x - 2) (x^{2} - 5)) = 0$

Remove unnecessary parentheses.

$3 (x - 2) (x^{2} - 5) = 0$

Step 9

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

$x - 2 = 0$

$x^{2} - 5 = 0$

Step 10

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 11

Set $x^{2} - 5$ equal to $0$ and solve for $x$ .

Set $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Solve $x^{2} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$x^{2} = 5$

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

$x = \pm \sqrt{5}$

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.

$x = \sqrt{5}$

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

$x = - \sqrt{5}$

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

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

Step 12

The final solution is all the values that make $3 (x - 2) (x^{2} - 5) = 0$ true.

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

Step 13

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2, 2.23606797 \dots, - 2.23606797 \dots$

Step 14

Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^3-6x^2-15x+30

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