Trigonometry Examples

$f (x) = 4 x^{4} - 35 x^{3} + 140 x^{2} - 295 x + 156$

Step 1

Regroup terms.

$f (x) = - 35 x^{3} + 140 x^{2} + 4 x^{4} - 295 x + 156$

Step 2

Factor $- 35 x^{2}$ out of $- 35 x^{3} + 140 x^{2}$ .

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Step 2.1

Factor $- 35 x^{2}$ out of $- 35 x^{3}$ .

$f (x) = - 35 x^{2} x + 140 x^{2} + 4 x^{4} - 295 x + 156$

Step 2.2

Factor $- 35 x^{2}$ out of $140 x^{2}$ .

$f (x) = - 35 x^{2} x - 35 x^{2} \cdot - 4 + 4 x^{4} - 295 x + 156$

Step 2.3

Factor $- 35 x^{2}$ out of $- 35 x^{2} (x) - 35 x^{2} (- 4)$ .

$f (x) = - 35 x^{2} (x - 4) + 4 x^{4} - 295 x + 156$

Step 3

Factor $4 x^{4} - 295 x + 156$ using the rational roots test.

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Step 3.1

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.

$p = \pm 1, \pm 156, \pm 2, \pm 78, \pm 3, \pm 52, \pm 4, \pm 39, \pm 6, \pm 26, \pm 12, \pm 13$

$q = \pm 1, \pm 4, \pm 2$

Step 3.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 156, \pm 39, \pm 78, \pm 2, \pm 19.5, \pm 3, \pm 0.75, \pm 1.5, \pm 52, \pm 13, \pm 26, \pm 4, \pm 9.75, \pm 6, \pm 6.5, \pm 12, \pm 3.25$

Step 3.3

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 3.3.1

Substitute $4$ into the polynomial.

$4 \cdot 4^{4} - 295 \cdot 4 + 156$

Step 3.3.2

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

$4 \cdot 256 - 295 \cdot 4 + 156$

Step 3.3.3

Multiply $4$ by $256$ .

$1024 - 295 \cdot 4 + 156$

Step 3.3.4

Multiply $- 295$ by $4$ .

$1024 - 1180 + 156$

Step 3.3.5

Subtract $1180$ from $1024$ .

$- 156 + 156$

Step 3.3.6

Add $- 156$ and $156$ .

$0$

Step 3.4

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.

$\frac{4 x^{4} - 295 x + 156}{x - 4}$

Step 3.5

Divide $4 x^{4} - 295 x + 156$ by $x - 4$ .

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Step 3.5.1

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

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

Step 3.5.2

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

$4 x^{3}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

Step 3.5.3

Multiply the new quotient term by the divisor.

$4 x^{3}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

Step 3.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{4} - 16 x^{3}$

$4 x^{3}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

Step 3.5.5

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

$4 x^{3}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

Step 3.5.6

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

$4 x^{3}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

Step 3.5.7

Divide the highest order term in the dividend $16 x^{3}$ by the highest order term in divisor $x$ .

$4 x^{3}$

$16 x^{2}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

Step 3.5.8

Multiply the new quotient term by the divisor.

$4 x^{3}$

$16 x^{2}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

Step 3.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{3} - 64 x^{2}$

$4 x^{3}$

$16 x^{2}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

Step 3.5.10

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

$4 x^{3}$

$16 x^{2}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

Step 3.5.11

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

$4 x^{3}$

$16 x^{2}$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

$295 x$

Step 3.5.12

Divide the highest order term in the dividend $64 x^{2}$ by the highest order term in divisor $x$ .

$4 x^{3}$

$16 x^{2}$

$64 x$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

$295 x$

Step 3.5.13

Multiply the new quotient term by the divisor.

$4 x^{3}$

$16 x^{2}$

$64 x$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

$295 x$

$64 x^{2}$

$256 x$

Step 3.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $64 x^{2} - 256 x$

$4 x^{3}$

$16 x^{2}$

$64 x$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

$295 x$

$64 x^{2}$

$256 x$

Step 3.5.15

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

$4 x^{3}$

$16 x^{2}$

$64 x$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

$295 x$

$64 x^{2}$

$256 x$

$39 x$

Step 3.5.16

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

$4 x^{3}$

$16 x^{2}$

$64 x$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

$295 x$

$64 x^{2}$

$256 x$

$39 x$

$156$

Step 3.5.17

Divide the highest order term in the dividend $- 39 x$ by the highest order term in divisor $x$ .

$4 x^{3}$

$16 x^{2}$

$64 x$

$39$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

$295 x$

$64 x^{2}$

$256 x$

$39 x$

$156$

Step 3.5.18

Multiply the new quotient term by the divisor.

$4 x^{3}$

$16 x^{2}$

$64 x$

$39$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

$295 x$

$64 x^{2}$

$256 x$

$39 x$

$156$

$39 x$

$156$

Step 3.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 39 x + 156$

$4 x^{3}$

$16 x^{2}$

$64 x$

$39$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

$295 x$

$64 x^{2}$

$256 x$

$39 x$

$156$

$39 x$

$156$

Step 3.5.20

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

$4 x^{3}$

$16 x^{2}$

$64 x$

$39$

$x$

$4$

$4 x^{4}$

$0 x^{3}$

$0 x^{2}$

$295 x$

$156$

$4 x^{4}$

$16 x^{3}$

$0 x^{2}$

$16 x^{3}$

$64 x^{2}$

$295 x$

$64 x^{2}$

$256 x$

$39 x$

$156$

$39 x$

$156$

$0$

Step 3.5.21

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

$4 x^{3} + 16 x^{2} + 64 x - 39$

Step 3.6

Write $4 x^{4} - 295 x + 156$ as a set of factors.

$f (x) = - 35 x^{2} (x - 4) + (x - 4) (4 x^{3} + 16 x^{2} + 64 x - 39)$

Step 4

Factor $x - 4$ out of $- 35 x^{2} (x - 4) + (x - 4) (4 x^{3} + 16 x^{2} + 64 x - 39)$ .

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Step 4.1

Factor $x - 4$ out of $- 35 x^{2} (x - 4)$ .

$f (x) = (x - 4) (- 35 x^{2}) + (x - 4) (4 x^{3} + 16 x^{2} + 64 x - 39)$

Step 4.2

Factor $x - 4$ out of $(x - 4) (- 35 x^{2}) + (x - 4) (4 x^{3} + 16 x^{2} + 64 x - 39)$ .

$f (x) = (x - 4) (- 35 x^{2} + 4 x^{3} + 16 x^{2} + 64 x - 39)$

Step 5

Add $- 35 x^{2}$ and $16 x^{2}$ .

$f (x) = (x - 4) (4 x^{3} - 19 x^{2} + 64 x - 39)$

Step 6

Factor.

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Step 6.1

Factor $4 x^{3} - 19 x^{2} + 64 x - 39$ using the rational roots test.

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Step 6.1.1

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.

$p = \pm 1, \pm 39, \pm 3, \pm 13$

$q = \pm 1, \pm 4, \pm 2$

Step 6.1.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 39, \pm 9.75, \pm 19.5, \pm 3, \pm 0.75, \pm 1.5, \pm 13, \pm 3.25, \pm 6.5$

Step 6.1.3

Substitute $0.75$ and simplify the expression. In this case, the expression is equal to $0$ so $0.75$ is a root of the polynomial.

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Step 6.1.3.1

Substitute $0.75$ into the polynomial.

$4 \cdot {0.75}^{3} - 19 \cdot {0.75}^{2} + 64 \cdot 0.75 - 39$

Step 6.1.3.2

Raise $0.75$ to the power of $3$ .

$4 \cdot 0.421875 - 19 \cdot {0.75}^{2} + 64 \cdot 0.75 - 39$

Step 6.1.3.3

Multiply $4$ by $0.421875$ .

$1.6875 - 19 \cdot {0.75}^{2} + 64 \cdot 0.75 - 39$

Step 6.1.3.4

Raise $0.75$ to the power of $2$ .

$1.6875 - 19 \cdot 0.5625 + 64 \cdot 0.75 - 39$

Step 6.1.3.5

Multiply $- 19$ by $0.5625$ .

$1.6875 - 10.6875 + 64 \cdot 0.75 - 39$

Step 6.1.3.6

Subtract $10.6875$ from $1.6875$ .

$- 9 + 64 \cdot 0.75 - 39$

Step 6.1.3.7

Multiply $64$ by $0.75$ .

$- 9 + 48 - 39$

Step 6.1.3.8

Add $- 9$ and $48$ .

$39 - 39$

Step 6.1.3.9

Subtract $39$ from $39$ .

$0$

Step 6.1.4

Since $0.75$ is a known root, divide the polynomial by $4 x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 19 x^{2} + 64 x - 39}{4 x - 3}$

Step 6.1.5

Divide $4 x^{3} - 19 x^{2} + 64 x - 39$ by $4 x - 3$ .

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Step 6.1.5.1

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

$4 x$

$3$

$4 x^{3}$

$19 x^{2}$

$64 x$

$39$

Step 6.1.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $4 x$ .

					$x^{2}$
	$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$

Step 6.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			+	$4 x^{3}$	-	$3 x^{2}$

Step 6.1.5.4

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

				$x^{2}$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$

Step 6.1.5.5

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

				$x^{2}$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$
					-	$16 x^{2}$

Step 6.1.5.6

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

				$x^{2}$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$
					-	$16 x^{2}$	+	$64 x$

Step 6.1.5.7

Divide the highest order term in the dividend $- 16 x^{2}$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$4 x$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$
					-	$16 x^{2}$	+	$64 x$

Step 6.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$
					-	$16 x^{2}$	+	$64 x$
					-	$16 x^{2}$	+	$12 x$

Step 6.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 16 x^{2} + 12 x$

				$x^{2}$	-	$4 x$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$
					-	$16 x^{2}$	+	$64 x$
					+	$16 x^{2}$	-	$12 x$

Step 6.1.5.10

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

				$x^{2}$	-	$4 x$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$
					-	$16 x^{2}$	+	$64 x$
					+	$16 x^{2}$	-	$12 x$
							+	$52 x$

Step 6.1.5.11

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

				$x^{2}$	-	$4 x$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$
					-	$16 x^{2}$	+	$64 x$
					+	$16 x^{2}$	-	$12 x$
							+	$52 x$	-	$39$

Step 6.1.5.12

Divide the highest order term in the dividend $52 x$ by the highest order term in divisor $4 x$ .

				$x^{2}$	-	$4 x$	+	$13$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$
					-	$16 x^{2}$	+	$64 x$
					+	$16 x^{2}$	-	$12 x$
							+	$52 x$	-	$39$

Step 6.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$13$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$
					-	$16 x^{2}$	+	$64 x$
					+	$16 x^{2}$	-	$12 x$
							+	$52 x$	-	$39$
							+	$52 x$	-	$39$

Step 6.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $52 x - 39$

				$x^{2}$	-	$4 x$	+	$13$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$
					-	$16 x^{2}$	+	$64 x$
					+	$16 x^{2}$	-	$12 x$
							+	$52 x$	-	$39$
							-	$52 x$	+	$39$

Step 6.1.5.15

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

				$x^{2}$	-	$4 x$	+	$13$
$4 x$	-	$3$		$4 x^{3}$	-	$19 x^{2}$	+	$64 x$	-	$39$
			-	$4 x^{3}$	+	$3 x^{2}$
					-	$16 x^{2}$	+	$64 x$
					+	$16 x^{2}$	-	$12 x$
							+	$52 x$	-	$39$
							-	$52 x$	+	$39$
										$0$

Step 6.1.5.16

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

$x^{2} - 4 x + 13$

Step 6.1.6

Write $4 x^{3} - 19 x^{2} + 64 x - 39$ as a set of factors.

$f (x) = (x - 4) ((4 x - 3) (x^{2} - 4 x + 13))$

Step 6.2

Remove unnecessary parentheses.

$f (x) = (x - 4) (4 x - 3) (x^{2} - 4 x + 13)$