Trigonometry Examples

$x^{4} + x^{3} - 54 x^{2} + 120 x = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $x$ out of $x^{4} + x^{3} - 54 x^{2} + 120 x$ .

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Step 1.1.1

Factor $x$ out of $x^{4}$ .

$x \cdot x^{3} + x^{3} - 54 x^{2} + 120 x = 0$

Step 1.1.2

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

$x \cdot x^{3} + x \cdot x^{2} - 54 x^{2} + 120 x = 0$

Step 1.1.3

Factor $x$ out of $- 54 x^{2}$ .

$x \cdot x^{3} + x \cdot x^{2} + x (- 54 x) + 120 x = 0$

Step 1.1.4

Factor $x$ out of $120 x$ .

$x \cdot x^{3} + x \cdot x^{2} + x (- 54 x) + x \cdot 120 = 0$

Step 1.1.5

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

$x (x^{3} + x^{2}) + x (- 54 x) + x \cdot 120 = 0$

Step 1.1.6

Factor $x$ out of $x (x^{3} + x^{2}) + x (- 54 x)$ .

$x (x^{3} + x^{2} - 54 x) + x \cdot 120 = 0$

Step 1.1.7

Factor $x$ out of $x (x^{3} + x^{2} - 54 x) + x \cdot 120$ .

$x (x^{3} + x^{2} - 54 x + 120) = 0$

Step 1.2

Factor.

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Step 1.2.1

Factor $x^{3} + x^{2} - 54 x + 120$ using the rational roots test.

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Step 1.2.1.1

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.

$p = \pm 1, \pm 120, \pm 2, \pm 60, \pm 3, \pm 40, \pm 4, \pm 30, \pm 5, \pm 24, \pm 6, \pm 20, \pm 8, \pm 15, \pm 10, \pm 12$

$q = \pm 1$

Step 1.2.1.2

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

$\pm 1, \pm 120, \pm 2, \pm 60, \pm 3, \pm 40, \pm 4, \pm 30, \pm 5, \pm 24, \pm 6, \pm 20, \pm 8, \pm 15, \pm 10, \pm 12$

Step 1.2.1.3

Substitute $5$ and simplify the expression. In this case, the expression is equal to $0$ so $5$ is a root of the polynomial.

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Step 1.2.1.3.1

Substitute $5$ into the polynomial.

$5^{3} + 5^{2} - 54 \cdot 5 + 120$

Step 1.2.1.3.2

Raise $5$ to the power of $3$ .

$125 + 5^{2} - 54 \cdot 5 + 120$

Step 1.2.1.3.3

Raise $5$ to the power of $2$ .

$125 + 25 - 54 \cdot 5 + 120$

Step 1.2.1.3.4

Add $125$ and $25$ .

$150 - 54 \cdot 5 + 120$

Step 1.2.1.3.5

Multiply $- 54$ by $5$ .

$150 - 270 + 120$

Step 1.2.1.3.6

Subtract $270$ from $150$ .

$- 120 + 120$

Step 1.2.1.3.7

Add $- 120$ and $120$ .

$0$

Step 1.2.1.4

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + x^{2} - 54 x + 120}{x - 5}$

Step 1.2.1.5

Divide $x^{3} + x^{2} - 54 x + 120$ by $x - 5$ .

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Step 1.2.1.5.1

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

$x$

$5$

$x^{3}$

$x^{2}$

$54 x$

$120$

Step 1.2.1.5.2

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

					$x^{2}$
	$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$

Step 1.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			+	$x^{3}$	-	$5 x^{2}$

Step 1.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 5 x^{2}$

				$x^{2}$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$

Step 1.2.1.5.5

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

				$x^{2}$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$
					+	$6 x^{2}$

Step 1.2.1.5.6

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

				$x^{2}$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$
					+	$6 x^{2}$	-	$54 x$

Step 1.2.1.5.7

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$6 x$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$
					+	$6 x^{2}$	-	$54 x$

Step 1.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$
					+	$6 x^{2}$	-	$54 x$
					+	$6 x^{2}$	-	$30 x$

Step 1.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} - 30 x$

				$x^{2}$	+	$6 x$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$
					+	$6 x^{2}$	-	$54 x$
					-	$6 x^{2}$	+	$30 x$

Step 1.2.1.5.10

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

				$x^{2}$	+	$6 x$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$
					+	$6 x^{2}$	-	$54 x$
					-	$6 x^{2}$	+	$30 x$
							-	$24 x$

Step 1.2.1.5.11

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

				$x^{2}$	+	$6 x$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$
					+	$6 x^{2}$	-	$54 x$
					-	$6 x^{2}$	+	$30 x$
							-	$24 x$	+	$120$

Step 1.2.1.5.12

Divide the highest order term in the dividend $- 24 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$6 x$	-	$24$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$
					+	$6 x^{2}$	-	$54 x$
					-	$6 x^{2}$	+	$30 x$
							-	$24 x$	+	$120$

Step 1.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$	-	$24$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$
					+	$6 x^{2}$	-	$54 x$
					-	$6 x^{2}$	+	$30 x$
							-	$24 x$	+	$120$
							-	$24 x$	+	$120$

Step 1.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x + 120$

				$x^{2}$	+	$6 x$	-	$24$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$
					+	$6 x^{2}$	-	$54 x$
					-	$6 x^{2}$	+	$30 x$
							-	$24 x$	+	$120$
							+	$24 x$	-	$120$

Step 1.2.1.5.15

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

				$x^{2}$	+	$6 x$	-	$24$
$x$	-	$5$		$x^{3}$	+	$x^{2}$	-	$54 x$	+	$120$
			-	$x^{3}$	+	$5 x^{2}$
					+	$6 x^{2}$	-	$54 x$
					-	$6 x^{2}$	+	$30 x$
							-	$24 x$	+	$120$
							+	$24 x$	-	$120$
										$0$

Step 1.2.1.5.16

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

$x^{2} + 6 x - 24$

Step 1.2.1.6

Write $x^{3} + x^{2} - 54 x + 120$ as a set of factors.

$x ((x - 5) (x^{2} + 6 x - 24)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$x (x - 5) (x^{2} + 6 x - 24) = 0$

Step 2

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

$x = 0$

$x - 5 = 0$

$x^{2} + 6 x - 24 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 5

Set $x^{2} + 6 x - 24$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} + 6 x - 24$ equal to $0$ .

$x^{2} + 6 x - 24 = 0$

Step 5.2

Solve $x^{2} + 6 x - 24 = 0$ for $x$ .

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Step 5.2.1

Use the quadratic formula to find the solutions.

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

Step 5.2.2

Substitute the values $a = 1$ , $b = 6$ , and $c = - 24$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot - 24)}}{2 \cdot 1}$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 24}}{2 \cdot 1}$

Step 5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 24$ .

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Step 5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 24}}{2 \cdot 1}$

Step 5.2.3.1.2.2

Multiply $- 4$ by $- 24$ .

$x = \frac{- 6 \pm \sqrt{36 + 96}}{2 \cdot 1}$

Step 5.2.3.1.3

Add $36$ and $96$ .

$x = \frac{- 6 \pm \sqrt{132}}{2 \cdot 1}$

Step 5.2.3.1.4

Rewrite $132$ as $2^{2} \cdot 33$ .

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Step 5.2.3.1.4.1

Factor $4$ out of $132$ .

$x = \frac{- 6 \pm \sqrt{4 (33)}}{2 \cdot 1}$

Step 5.2.3.1.4.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 33}}{2 \cdot 1}$

Step 5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{33}}{2 \cdot 1}$

Step 5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 \sqrt{33}}{2}$

Step 5.2.3.3

Simplify $\frac{- 6 \pm 2 \sqrt{33}}{2}$ .

$x = - 3 \pm \sqrt{33}$

Step 5.2.4

The final answer is the combination of both solutions.

$x = - 3 + \sqrt{33}, - 3 - \sqrt{33}$

Step 6

The final solution is all the values that make $x (x - 5) (x^{2} + 6 x - 24) = 0$ true.

$x = 0, 5, - 3 + \sqrt{33}, - 3 - \sqrt{33}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 0, 5, - 3 + \sqrt{33}, - 3 - \sqrt{33}$

Decimal Form:

$x = 0, 5, 2.74456264 \dots, - 8.74456264 \dots$

Step 8