Trigonometry Examples

$f (x) = 3 x^{3} - 33 x^{2} + 60 x + 96$

Step 1

Factor $3$ out of $3 x^{3} - 33 x^{2} + 60 x + 96$ .

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Step 1.1

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

$f (x) = 3 (x^{3}) - 33 x^{2} + 60 x + 96$

Step 1.2

Factor $3$ out of $- 33 x^{2}$ .

$f (x) = 3 (x^{3}) + 3 (- 11 x^{2}) + 60 x + 96$

Step 1.3

Factor $3$ out of $60 x$ .

$f (x) = 3 (x^{3}) + 3 (- 11 x^{2}) + 3 (20 x) + 96$

Step 1.4

Factor $3$ out of $96$ .

$f (x) = 3 x^{3} + 3 (- 11 x^{2}) + 3 (20 x) + 3 \cdot 32$

Step 1.5

Factor $3$ out of $3 x^{3} + 3 (- 11 x^{2})$ .

$f (x) = 3 (x^{3} - 11 x^{2}) + 3 (20 x) + 3 \cdot 32$

Step 1.6

Factor $3$ out of $3 (x^{3} - 11 x^{2}) + 3 (20 x)$ .

$f (x) = 3 (x^{3} - 11 x^{2} + 20 x) + 3 \cdot 32$

Step 1.7

Factor $3$ out of $3 (x^{3} - 11 x^{2} + 20 x) + 3 \cdot 32$ .

$f (x) = 3 (x^{3} - 11 x^{2} + 20 x + 32)$

Step 2

Factor $x^{3} - 11 x^{2} + 20 x + 32$ using the rational roots test.

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Step 2.1

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.

$p = \pm 1, \pm 32, \pm 2, \pm 16, \pm 4, \pm 8$

$q = \pm 1$

Step 2.2

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

$\pm 1, \pm 32, \pm 2, \pm 16, \pm 4, \pm 8$

Step 2.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 2.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - 11 {(- 1)}^{2} + 20 \cdot - 1 + 32$

Step 2.3.2

Raise $- 1$ to the power of $3$ .

$- 1 - 11 {(- 1)}^{2} + 20 \cdot - 1 + 32$

Step 2.3.3

Raise $- 1$ to the power of $2$ .

$- 1 - 11 \cdot 1 + 20 \cdot - 1 + 32$

Step 2.3.4

Multiply $- 11$ by $1$ .

$- 1 - 11 + 20 \cdot - 1 + 32$

Step 2.3.5

Subtract $11$ from $- 1$ .

$- 12 + 20 \cdot - 1 + 32$

Step 2.3.6

Multiply $20$ by $- 1$ .

$- 12 - 20 + 32$

Step 2.3.7

Subtract $20$ from $- 12$ .

$- 32 + 32$

Step 2.3.8

Add $- 32$ and $32$ .

$0$

Step 2.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 11 x^{2} + 20 x + 32}{x + 1}$

Step 2.5

Divide $x^{3} - 11 x^{2} + 20 x + 32$ by $x + 1$ .

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Step 2.5.1

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

$x$

$1$

$x^{3}$

$11 x^{2}$

$20 x$

$32$

Step 2.5.2

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$

Step 2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			+	$x^{3}$	+	$x^{2}$

Step 2.5.4

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$

Step 2.5.5

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$
					-	$12 x^{2}$

Step 2.5.6

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$
					-	$12 x^{2}$	+	$20 x$

Step 2.5.7

Divide the highest order term in the dividend $- 12 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$12 x$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$
					-	$12 x^{2}$	+	$20 x$

Step 2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$12 x$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$
					-	$12 x^{2}$	+	$20 x$
					-	$12 x^{2}$	-	$12 x$

Step 2.5.9

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

				$x^{2}$	-	$12 x$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$
					-	$12 x^{2}$	+	$20 x$
					+	$12 x^{2}$	+	$12 x$

Step 2.5.10

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

				$x^{2}$	-	$12 x$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$
					-	$12 x^{2}$	+	$20 x$
					+	$12 x^{2}$	+	$12 x$
							+	$32 x$

Step 2.5.11

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

				$x^{2}$	-	$12 x$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$
					-	$12 x^{2}$	+	$20 x$
					+	$12 x^{2}$	+	$12 x$
							+	$32 x$	+	$32$

Step 2.5.12

Divide the highest order term in the dividend $32 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$12 x$	+	$32$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$
					-	$12 x^{2}$	+	$20 x$
					+	$12 x^{2}$	+	$12 x$
							+	$32 x$	+	$32$

Step 2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$12 x$	+	$32$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$
					-	$12 x^{2}$	+	$20 x$
					+	$12 x^{2}$	+	$12 x$
							+	$32 x$	+	$32$
							+	$32 x$	+	$32$

Step 2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $32 x + 32$

				$x^{2}$	-	$12 x$	+	$32$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$
					-	$12 x^{2}$	+	$20 x$
					+	$12 x^{2}$	+	$12 x$
							+	$32 x$	+	$32$
							-	$32 x$	-	$32$

Step 2.5.15

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

				$x^{2}$	-	$12 x$	+	$32$
$x$	+	$1$		$x^{3}$	-	$11 x^{2}$	+	$20 x$	+	$32$
			-	$x^{3}$	-	$x^{2}$
					-	$12 x^{2}$	+	$20 x$
					+	$12 x^{2}$	+	$12 x$
							+	$32 x$	+	$32$
							-	$32 x$	-	$32$
										$0$

Step 2.5.16

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

$x^{2} - 12 x + 32$

Step 2.6

Write $x^{3} - 11 x^{2} + 20 x + 32$ as a set of factors.

$f (x) = 3 ((x + 1) (x^{2} - 12 x + 32))$

Step 3

Factor.

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Step 3.1

Factor $x^{2} - 12 x + 32$ using the AC method.

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Step 3.1.1

Factor $x^{2} - 12 x + 32$ using the AC method.

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Step 3.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $32$ and whose sum is $- 12$ .

$- 8, - 4$

Step 3.1.1.2

Write the factored form using these integers.

$f (x) = 3 ((x + 1) ((x - 8) (x - 4)))$

Step 3.1.2

Remove unnecessary parentheses.

$f (x) = 3 ((x + 1) (x - 8) (x - 4))$

Step 3.2

Remove unnecessary parentheses.

$f (x) = 3 (x + 1) (x - 8) (x - 4)$