Trigonometry Examples

$f (x) = 2 x^{4} - 19 x^{3} + 33 x^{2} + 59 x - 60$

Step 1

Factor $2 x^{4} - 19 x^{3} + 33 x^{2} + 59 x - 60$ using the rational roots test.

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Step 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.

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

$q = \pm 1, \pm 2$

Step 1.2

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

$\pm 1, \pm 0.5, \pm 60, \pm 30, \pm 2, \pm 15, \pm 3, \pm 1.5, \pm 20, \pm 10, \pm 4, \pm 7.5, \pm 5, \pm 2.5, \pm 12, \pm 6$

Step 1.3

Substitute $- 1.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1.5$ is a root of the polynomial.

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Step 1.3.1

Substitute $- 1.5$ into the polynomial.

$2 {(- 1.5)}^{4} - 19 {(- 1.5)}^{3} + 33 {(- 1.5)}^{2} + 59 \cdot - 1.5 - 60$

Step 1.3.2

Raise $- 1.5$ to the power of $4$ .

$2 \cdot 5.0625 - 19 {(- 1.5)}^{3} + 33 {(- 1.5)}^{2} + 59 \cdot - 1.5 - 60$

Step 1.3.3

Multiply $2$ by $5.0625$ .

$10.125 - 19 {(- 1.5)}^{3} + 33 {(- 1.5)}^{2} + 59 \cdot - 1.5 - 60$

Step 1.3.4

Raise $- 1.5$ to the power of $3$ .

$10.125 - 19 \cdot - 3.375 + 33 {(- 1.5)}^{2} + 59 \cdot - 1.5 - 60$

Step 1.3.5

Multiply $- 19$ by $- 3.375$ .

$10.125 + 64.125 + 33 {(- 1.5)}^{2} + 59 \cdot - 1.5 - 60$

Step 1.3.6

Add $10.125$ and $64.125$ .

$74.25 + 33 {(- 1.5)}^{2} + 59 \cdot - 1.5 - 60$

Step 1.3.7

Raise $- 1.5$ to the power of $2$ .

$74.25 + 33 \cdot 2.25 + 59 \cdot - 1.5 - 60$

Step 1.3.8

Multiply $33$ by $2.25$ .

$74.25 + 74.25 + 59 \cdot - 1.5 - 60$

Step 1.3.9

Add $74.25$ and $74.25$ .

$148.5 + 59 \cdot - 1.5 - 60$

Step 1.3.10

Multiply $59$ by $- 1.5$ .

$148.5 - 88.5 - 60$

Step 1.3.11

Subtract $88.5$ from $148.5$ .

$60 - 60$

Step 1.3.12

Subtract $60$ from $60$ .

$0$

Step 1.4

Since $- 1.5$ is a known root, divide the polynomial by $2 x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{4} - 19 x^{3} + 33 x^{2} + 59 x - 60}{2 x + 3}$

Step 1.5

Divide $2 x^{4} - 19 x^{3} + 33 x^{2} + 59 x - 60$ by $2 x + 3$ .

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Step 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 $0$ .

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

Step 1.5.2

Divide the highest order term in the dividend $2 x^{4}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

Step 1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} + 3 x^{3}$

$x^{3}$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

Step 1.5.5

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

$x^{3}$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

Step 1.5.6

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

$x^{3}$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

Step 1.5.7

Divide the highest order term in the dividend $- 22 x^{3}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$11 x^{2}$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

Step 1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$11 x^{2}$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 22 x^{3} - 33 x^{2}$

$x^{3}$

$11 x^{2}$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

Step 1.5.10

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

$x^{3}$

$11 x^{2}$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

$66 x^{2}$

Step 1.5.11

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

$x^{3}$

$11 x^{2}$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

$66 x^{2}$

$59 x$

Step 1.5.12

Divide the highest order term in the dividend $66 x^{2}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$11 x^{2}$

$33 x$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

$66 x^{2}$

$59 x$

Step 1.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$11 x^{2}$

$33 x$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

$66 x^{2}$

$59 x$

$66 x^{2}$

$99 x$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $66 x^{2} + 99 x$

$x^{3}$

$11 x^{2}$

$33 x$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

$66 x^{2}$

$59 x$

$66 x^{2}$

$99 x$

Step 1.5.15

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

$x^{3}$

$11 x^{2}$

$33 x$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

$66 x^{2}$

$59 x$

$66 x^{2}$

$99 x$

$40 x$

Step 1.5.16

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

$x^{3}$

$11 x^{2}$

$33 x$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

$66 x^{2}$

$59 x$

$66 x^{2}$

$99 x$

$40 x$

$60$

Step 1.5.17

Divide the highest order term in the dividend $- 40 x$ by the highest order term in divisor $2 x$ .

$x^{3}$

$11 x^{2}$

$33 x$

$20$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

$66 x^{2}$

$59 x$

$66 x^{2}$

$99 x$

$40 x$

$60$

Step 1.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$11 x^{2}$

$33 x$

$20$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

$66 x^{2}$

$59 x$

$66 x^{2}$

$99 x$

$40 x$

$60$

$40 x$

$60$

Step 1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 40 x - 60$

$x^{3}$

$11 x^{2}$

$33 x$

$20$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

$66 x^{2}$

$59 x$

$66 x^{2}$

$99 x$

$40 x$

$60$

$40 x$

$60$

Step 1.5.20

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

$x^{3}$

$11 x^{2}$

$33 x$

$20$

$2 x$

$3$

$2 x^{4}$

$19 x^{3}$

$33 x^{2}$

$59 x$

$60$

$2 x^{4}$

$3 x^{3}$

$22 x^{3}$

$33 x^{2}$

$22 x^{3}$

$33 x^{2}$

$66 x^{2}$

$59 x$

$66 x^{2}$

$99 x$

$40 x$

$60$

$40 x$

$60$

$0$

Step 1.5.21

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

$x^{3} - 11 x^{2} + 33 x - 20$

Step 1.6

Write $2 x^{4} - 19 x^{3} + 33 x^{2} + 59 x - 60$ as a set of factors.

$f (x) = (2 x + 3) (x^{3} - 11 x^{2} + 33 x - 20)$

Step 2

Factor $x^{3} - 11 x^{2} + 33 x - 20$ using the rational roots test.

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Step 2.1

Factor $x^{3} - 11 x^{2} + 33 x - 20$ 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.

$p = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

$q = \pm 1$

Step 2.1.2

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

$\pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

Step 2.1.3

Substitute $4$ and simplify the expression. In this case, the expression is equal to $0$ so $4$ is a root of the polynomial.

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Step 2.1.3.1

Substitute $4$ into the polynomial.

$4^{3} - 11 \cdot 4^{2} + 33 \cdot 4 - 20$

Step 2.1.3.2

Raise $4$ to the power of $3$ .

$64 - 11 \cdot 4^{2} + 33 \cdot 4 - 20$

Step 2.1.3.3

Raise $4$ to the power of $2$ .

$64 - 11 \cdot 16 + 33 \cdot 4 - 20$

Step 2.1.3.4

Multiply $- 11$ by $16$ .

$64 - 176 + 33 \cdot 4 - 20$

Step 2.1.3.5

Subtract $176$ from $64$ .

$- 112 + 33 \cdot 4 - 20$

Step 2.1.3.6

Multiply $33$ by $4$ .

$- 112 + 132 - 20$

Step 2.1.3.7

Add $- 112$ and $132$ .

$20 - 20$

Step 2.1.3.8

Subtract $20$ from $20$ .

$0$

Step 2.1.4

Since $4$ is a known root, divide the polynomial by $x - 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 11 x^{2} + 33 x - 20}{x - 4}$

Step 2.1.5

Divide $x^{3} - 11 x^{2} + 33 x - 20$ by $x - 4$ .

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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 $0$ .

$x$

$4$

$x^{3}$

$11 x^{2}$

$33 x$

$20$

Step 2.1.5.2

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

					$x^{2}$
	$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			+	$x^{3}$	-	$4 x^{2}$

Step 2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 4 x^{2}$

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$

Step 2.1.5.5

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$
					-	$7 x^{2}$

Step 2.1.5.6

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$
					-	$7 x^{2}$	+	$33 x$

Step 2.1.5.7

Divide the highest order term in the dividend $- 7 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$7 x$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$
					-	$7 x^{2}$	+	$33 x$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$7 x$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$
					-	$7 x^{2}$	+	$33 x$
					-	$7 x^{2}$	+	$28 x$

Step 2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{2} + 28 x$

				$x^{2}$	-	$7 x$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$
					-	$7 x^{2}$	+	$33 x$
					+	$7 x^{2}$	-	$28 x$

Step 2.1.5.10

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

				$x^{2}$	-	$7 x$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$
					-	$7 x^{2}$	+	$33 x$
					+	$7 x^{2}$	-	$28 x$
							+	$5 x$

Step 2.1.5.11

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

				$x^{2}$	-	$7 x$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$
					-	$7 x^{2}$	+	$33 x$
					+	$7 x^{2}$	-	$28 x$
							+	$5 x$	-	$20$

Step 2.1.5.12

Divide the highest order term in the dividend $5 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$7 x$	+	$5$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$
					-	$7 x^{2}$	+	$33 x$
					+	$7 x^{2}$	-	$28 x$
							+	$5 x$	-	$20$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$7 x$	+	$5$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$
					-	$7 x^{2}$	+	$33 x$
					+	$7 x^{2}$	-	$28 x$
							+	$5 x$	-	$20$
							+	$5 x$	-	$20$

Step 2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $5 x - 20$

				$x^{2}$	-	$7 x$	+	$5$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$
					-	$7 x^{2}$	+	$33 x$
					+	$7 x^{2}$	-	$28 x$
							+	$5 x$	-	$20$
							-	$5 x$	+	$20$

Step 2.1.5.15

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

				$x^{2}$	-	$7 x$	+	$5$
$x$	-	$4$		$x^{3}$	-	$11 x^{2}$	+	$33 x$	-	$20$
			-	$x^{3}$	+	$4 x^{2}$
					-	$7 x^{2}$	+	$33 x$
					+	$7 x^{2}$	-	$28 x$
							+	$5 x$	-	$20$
							-	$5 x$	+	$20$
										$0$

Step 2.1.5.16

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

$x^{2} - 7 x + 5$

Step 2.1.6

Write $x^{3} - 11 x^{2} + 33 x - 20$ as a set of factors.

$f (x) = (2 x + 3) ((x - 4) (x^{2} - 7 x + 5))$

Step 2.2

Remove unnecessary parentheses.

$f (x) = (2 x + 3) (x - 4) (x^{2} - 7 x + 5)$