Trigonometry Examples

$f (x) = - x^{4} + 11 x^{3} - 42 x^{2} + 60 x$

Step 1

Factor $x$ out of $- x^{4} + 11 x^{3} - 42 x^{2} + 60 x$ .

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Step 1.1

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

$f (x) = x (- x^{3}) + 11 x^{3} - 42 x^{2} + 60 x$

Step 1.2

Factor $x$ out of $11 x^{3}$ .

$f (x) = x (- x^{3}) + x (11 x^{2}) - 42 x^{2} + 60 x$

Step 1.3

Factor $x$ out of $- 42 x^{2}$ .

$f (x) = x (- x^{3}) + x (11 x^{2}) + x (- 42 x) + 60 x$

Step 1.4

Factor $x$ out of $60 x$ .

$f (x) = x (- x^{3}) + x (11 x^{2}) + x (- 42 x) + x \cdot 60$

Step 1.5

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

$f (x) = x (- x^{3} + 11 x^{2}) + x (- 42 x) + x \cdot 60$

Step 1.6

Factor $x$ out of $x (- x^{3} + 11 x^{2}) + x (- 42 x)$ .

$f (x) = x (- x^{3} + 11 x^{2} - 42 x) + x \cdot 60$

Step 1.7

Factor $x$ out of $x (- x^{3} + 11 x^{2} - 42 x) + x \cdot 60$ .

$f (x) = x (- x^{3} + 11 x^{2} - 42 x + 60)$

Step 2

Factor.

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Step 2.1

Factor $- x^{3} + 11 x^{2} - 42 x + 60$ 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.

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

$q = \pm 1$

Step 2.1.2

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

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

Step 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 2.1.3.1

Substitute $5$ into the polynomial.

$- 5^{3} + 11 \cdot 5^{2} - 42 \cdot 5 + 60$

Step 2.1.3.2

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

$- 1 \cdot 125 + 11 \cdot 5^{2} - 42 \cdot 5 + 60$

Step 2.1.3.3

Multiply $- 1$ by $125$ .

$- 125 + 11 \cdot 5^{2} - 42 \cdot 5 + 60$

Step 2.1.3.4

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

$- 125 + 11 \cdot 25 - 42 \cdot 5 + 60$

Step 2.1.3.5

Multiply $11$ by $25$ .

$- 125 + 275 - 42 \cdot 5 + 60$

Step 2.1.3.6

Add $- 125$ and $275$ .

$150 - 42 \cdot 5 + 60$

Step 2.1.3.7

Multiply $- 42$ by $5$ .

$150 - 210 + 60$

Step 2.1.3.8

Subtract $210$ from $150$ .

$- 60 + 60$

Step 2.1.3.9

Add $- 60$ and $60$ .

$0$

Step 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.

$\frac{- x^{3} + 11 x^{2} - 42 x + 60}{x - 5}$

Step 2.1.5

Divide $- x^{3} + 11 x^{2} - 42 x + 60$ by $x - 5$ .

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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 $0$ .

$x$

$5$

$x^{3}$

$11 x^{2}$

$42 x$

$60$

Step 2.1.5.2

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

				-	$x^{2}$
	$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			-	$x^{3}$	+	$5 x^{2}$

Step 2.1.5.4

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

			-	$x^{2}$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$

Step 2.1.5.5

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

			-	$x^{2}$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$
					+	$6 x^{2}$

Step 2.1.5.6

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

			-	$x^{2}$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$
					+	$6 x^{2}$	-	$42 x$

Step 2.1.5.7

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

			-	$x^{2}$	+	$6 x$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$
					+	$6 x^{2}$	-	$42 x$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$6 x$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$
					+	$6 x^{2}$	-	$42 x$
					+	$6 x^{2}$	-	$30 x$

Step 2.1.5.9

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

			-	$x^{2}$	+	$6 x$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$
					+	$6 x^{2}$	-	$42 x$
					-	$6 x^{2}$	+	$30 x$

Step 2.1.5.10

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

			-	$x^{2}$	+	$6 x$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$
					+	$6 x^{2}$	-	$42 x$
					-	$6 x^{2}$	+	$30 x$
							-	$12 x$

Step 2.1.5.11

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

			-	$x^{2}$	+	$6 x$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$
					+	$6 x^{2}$	-	$42 x$
					-	$6 x^{2}$	+	$30 x$
							-	$12 x$	+	$60$

Step 2.1.5.12

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $x$ .

			-	$x^{2}$	+	$6 x$	-	$12$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$
					+	$6 x^{2}$	-	$42 x$
					-	$6 x^{2}$	+	$30 x$
							-	$12 x$	+	$60$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$6 x$	-	$12$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$
					+	$6 x^{2}$	-	$42 x$
					-	$6 x^{2}$	+	$30 x$
							-	$12 x$	+	$60$
							-	$12 x$	+	$60$

Step 2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x + 60$

			-	$x^{2}$	+	$6 x$	-	$12$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$
					+	$6 x^{2}$	-	$42 x$
					-	$6 x^{2}$	+	$30 x$
							-	$12 x$	+	$60$
							+	$12 x$	-	$60$

Step 2.1.5.15

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

			-	$x^{2}$	+	$6 x$	-	$12$
$x$	-	$5$	-	$x^{3}$	+	$11 x^{2}$	-	$42 x$	+	$60$
			+	$x^{3}$	-	$5 x^{2}$
					+	$6 x^{2}$	-	$42 x$
					-	$6 x^{2}$	+	$30 x$
							-	$12 x$	+	$60$
							+	$12 x$	-	$60$
										$0$

Step 2.1.5.16

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

$- x^{2} + 6 x - 12$

Step 2.1.6

Write $- x^{3} + 11 x^{2} - 42 x + 60$ as a set of factors.

$f (x) = x ((x - 5) (- x^{2} + 6 x - 12))$

Step 2.2

Remove unnecessary parentheses.

$f (x) = x (x - 5) (- x^{2} + 6 x - 12)$