Trigonometry Examples

$f (x) = 3 x^{3} - 16 x^{2} - 30 x - 12$

Step 1

Set $3 x^{3} - 16 x^{2} - 30 x - 12$ equal to $0$ .

$3 x^{3} - 16 x^{2} - 30 x - 12 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $3 x^{3} - 16 x^{2} - 30 x - 12$ using the rational roots test.

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Step 2.1.1

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.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

$q = \pm 1, \pm 3$

Step 2.1.2

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

$\pm 1, \pm 0.\overline{3}, \pm 12, \pm 4, \pm 2, \pm 0.\overline{6}, \pm 6, \pm 3, \pm 1.\overline{3}$

Step 2.1.3

Substitute $- 0.\overline{6}$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.\overline{6}$ is a root of the polynomial.

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Step 2.1.3.1

Substitute $- 0.\overline{6}$ into the polynomial.

$3 {(- 0.\overline{6})}^{3} - 16 {(- 0.\overline{6})}^{2} - 30 \cdot - 0.\overline{6} - 12$

Step 2.1.3.2

Raise $- 0.\overline{6}$ to the power of $3$ .

$3 \cdot - 0.\overline{296} - 16 {(- 0.\overline{6})}^{2} - 30 \cdot - 0.\overline{6} - 12$

Step 2.1.3.3

Multiply $3$ by $- 0.\overline{296}$ .

$- 0.\overline{8} - 16 {(- 0.\overline{6})}^{2} - 30 \cdot - 0.\overline{6} - 12$

Step 2.1.3.4

Raise $- 0.\overline{6}$ to the power of $2$ .

$- 0.\overline{8} - 16 \cdot 0.\overline{4} - 30 \cdot - 0.\overline{6} - 12$

Step 2.1.3.5

Multiply $- 16$ by $0.\overline{4}$ .

$- 0.\overline{8} - 7.11111111 - 30 \cdot - 0.\overline{6} - 12$

Step 2.1.3.6

Subtract $7.11111111$ from $- 0.\overline{8}$ .

$- 8 - 30 \cdot - 0.\overline{6} - 12$

Step 2.1.3.7

Multiply $- 30$ by $- 0.\overline{6}$ .

$- 8 + 20 - 12$

Step 2.1.3.8

Add $- 8$ and $20$ .

$12 - 12$

Step 2.1.3.9

Subtract $12$ from $12$ .

$0$

Step 2.1.4

Since $- 0.\overline{6}$ is a known root, divide the polynomial by $3 x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{3} - 16 x^{2} - 30 x - 12}{3 x + 2}$

Step 2.1.5

Divide $3 x^{3} - 16 x^{2} - 30 x - 12$ by $3 x + 2$ .

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Step 2.1.5.1

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

$3 x$

$2$

$3 x^{3}$

$16 x^{2}$

$30 x$

$12$

Step 2.1.5.2

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

					$x^{2}$
	$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			+	$3 x^{3}$	+	$2 x^{2}$

Step 2.1.5.4

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

				$x^{2}$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$

Step 2.1.5.5

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

				$x^{2}$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$18 x^{2}$

Step 2.1.5.6

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

				$x^{2}$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$18 x^{2}$	-	$30 x$

Step 2.1.5.7

Divide the highest order term in the dividend $- 18 x^{2}$ by the highest order term in divisor $3 x$ .

				$x^{2}$	-	$6 x$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$18 x^{2}$	-	$30 x$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$18 x^{2}$	-	$30 x$
					-	$18 x^{2}$	-	$12 x$

Step 2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{2} - 12 x$

				$x^{2}$	-	$6 x$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$18 x^{2}$	-	$30 x$
					+	$18 x^{2}$	+	$12 x$

Step 2.1.5.10

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

				$x^{2}$	-	$6 x$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$18 x^{2}$	-	$30 x$
					+	$18 x^{2}$	+	$12 x$
							-	$18 x$

Step 2.1.5.11

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

				$x^{2}$	-	$6 x$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$18 x^{2}$	-	$30 x$
					+	$18 x^{2}$	+	$12 x$
							-	$18 x$	-	$12$

Step 2.1.5.12

Divide the highest order term in the dividend $- 18 x$ by the highest order term in divisor $3 x$ .

				$x^{2}$	-	$6 x$	-	$6$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$18 x^{2}$	-	$30 x$
					+	$18 x^{2}$	+	$12 x$
							-	$18 x$	-	$12$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	-	$6$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$18 x^{2}$	-	$30 x$
					+	$18 x^{2}$	+	$12 x$
							-	$18 x$	-	$12$
							-	$18 x$	-	$12$

Step 2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x - 12$

				$x^{2}$	-	$6 x$	-	$6$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$18 x^{2}$	-	$30 x$
					+	$18 x^{2}$	+	$12 x$
							-	$18 x$	-	$12$
							+	$18 x$	+	$12$

Step 2.1.5.15

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

				$x^{2}$	-	$6 x$	-	$6$
$3 x$	+	$2$		$3 x^{3}$	-	$16 x^{2}$	-	$30 x$	-	$12$
			-	$3 x^{3}$	-	$2 x^{2}$
					-	$18 x^{2}$	-	$30 x$
					+	$18 x^{2}$	+	$12 x$
							-	$18 x$	-	$12$
							+	$18 x$	+	$12$
										$0$

Step 2.1.5.16

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

$x^{2} - 6 x - 6$

Step 2.1.6

Write $3 x^{3} - 16 x^{2} - 30 x - 12$ as a set of factors.

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

Step 2.2

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

$3 x + 2 = 0$

$x^{2} - 6 x - 6 = 0$

Step 2.3

Set $3 x + 2$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $3 x + 2$ equal to $0$ .

$3 x + 2 = 0$

Step 2.3.2

Solve $3 x + 2 = 0$ for $x$ .

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Step 2.3.2.1

Subtract $2$ from both sides of the equation.

$3 x = - 2$

Step 2.3.2.2

Divide each term in $3 x = - 2$ by $3$ and simplify.

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Step 2.3.2.2.1

Divide each term in $3 x = - 2$ by $3$ .

$\frac{3 x}{3} = \frac{- 2}{3}$

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 2}{3}$

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{3}$

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

Step 2.4

Set $x^{2} - 6 x - 6$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2} - 6 x - 6$ equal to $0$ .

$x^{2} - 6 x - 6 = 0$

Step 2.4.2

Solve $x^{2} - 6 x - 6 = 0$ for $x$ .

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Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 1$ , $b = - 6$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot - 6)}}{2 \cdot 1}$

Step 2.4.2.3

Simplify.

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Step 2.4.2.3.1

Simplify the numerator.

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Step 2.4.2.3.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Step 2.4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 6$ .

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 6}}{2 \cdot 1}$

Step 2.4.2.3.1.2.2

Multiply $- 4$ by $- 6$ .

$x = \frac{6 \pm \sqrt{36 + 24}}{2 \cdot 1}$

Step 2.4.2.3.1.3

Add $36$ and $24$ .

$x = \frac{6 \pm \sqrt{60}}{2 \cdot 1}$

Step 2.4.2.3.1.4

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 2.4.2.3.1.4.1

Factor $4$ out of $60$ .

$x = \frac{6 \pm \sqrt{4 (15)}}{2 \cdot 1}$

Step 2.4.2.3.1.4.2

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot 1}$

Step 2.4.2.3.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{15}}{2 \cdot 1}$

Step 2.4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{15}}{2}$

Step 2.4.2.3.3

Simplify $\frac{6 \pm 2 \sqrt{15}}{2}$ .

$x = 3 \pm \sqrt{15}$

Step 2.4.2.4

The final answer is the combination of both solutions.

$x = 3 + \sqrt{15}, 3 - \sqrt{15}$

Step 2.5

The final solution is all the values that make $(3 x + 2) (x^{2} - 6 x - 6) = 0$ true.

$x = - \frac{2}{3}, 3 + \sqrt{15}, 3 - \sqrt{15}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2}{3}, 3 + \sqrt{15}, 3 - \sqrt{15}$

Decimal Form:

$x = - 0.\overline{6}, 6.87298334 \dots, - 0.87298334 \dots$

Step 4