Trigonometry Examples

$2 x^{2} + 7 x^{3} - 8 x^{2} + 5$

Step 1

Regroup terms.

$- 8 x^{2} + 2 x^{2} + 7 x^{3} + 5$

Step 2

Reorder terms.

$- 8 x^{2} + 7 x^{3} + 2 x^{2} + 5$

Step 3

Factor $7 x^{3} + 2 x^{2} + 5$ 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.

$p = \pm 1, \pm 5$

$q = \pm 1, \pm 7$

Step 3.2

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

$\pm 1, \pm 0.\overline{142857}, \pm 5, \pm 0.\overline{714285}$

Step 3.3

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 3.3.1

Substitute $- 1$ into the polynomial.

$7 {(- 1)}^{3} + 2 {(- 1)}^{2} + 5$

Step 3.3.2

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

$7 \cdot - 1 + 2 {(- 1)}^{2} + 5$

Step 3.3.3

Multiply $7$ by $- 1$ .

$- 7 + 2 {(- 1)}^{2} + 5$

Step 3.3.4

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

$- 7 + 2 \cdot 1 + 5$

Step 3.3.5

Multiply $2$ by $1$ .

$- 7 + 2 + 5$

Step 3.3.6

Add $- 7$ and $2$ .

$- 5 + 5$

Step 3.3.7

Add $- 5$ and $5$ .

$0$

Step 3.4

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.

$\frac{7 x^{3} + 2 x^{2} + 5}{x + 1}$

Step 3.5

Divide $7 x^{3} + 2 x^{2} + 5$ by $x + 1$ .

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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 $0$ .

$x$

$1$

$7 x^{3}$

$2 x^{2}$

$0 x$

$5$

Step 3.5.2

Divide the highest order term in the dividend $7 x^{3}$ by the highest order term in divisor $x$ .

					$7 x^{2}$
	$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$

Step 3.5.3

Multiply the new quotient term by the divisor.

				$7 x^{2}$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			+	$7 x^{3}$	+	$7 x^{2}$

Step 3.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{3} + 7 x^{2}$

				$7 x^{2}$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$

Step 3.5.5

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

				$7 x^{2}$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$
					-	$5 x^{2}$

Step 3.5.6

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

				$7 x^{2}$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$
					-	$5 x^{2}$	+	$0 x$

Step 3.5.7

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

				$7 x^{2}$	-	$5 x$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$
					-	$5 x^{2}$	+	$0 x$

Step 3.5.8

Multiply the new quotient term by the divisor.

				$7 x^{2}$	-	$5 x$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					-	$5 x^{2}$	-	$5 x$

Step 3.5.9

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

				$7 x^{2}$	-	$5 x$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	+	$5 x$

Step 3.5.10

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

				$7 x^{2}$	-	$5 x$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	+	$5 x$
							+	$5 x$

Step 3.5.11

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

				$7 x^{2}$	-	$5 x$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	+	$5 x$
							+	$5 x$	+	$5$

Step 3.5.12

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

				$7 x^{2}$	-	$5 x$	+	$5$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	+	$5 x$
							+	$5 x$	+	$5$

Step 3.5.13

Multiply the new quotient term by the divisor.

				$7 x^{2}$	-	$5 x$	+	$5$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	+	$5 x$
							+	$5 x$	+	$5$
							+	$5 x$	+	$5$

Step 3.5.14

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

				$7 x^{2}$	-	$5 x$	+	$5$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	+	$5 x$
							+	$5 x$	+	$5$
							-	$5 x$	-	$5$

Step 3.5.15

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

				$7 x^{2}$	-	$5 x$	+	$5$
$x$	+	$1$		$7 x^{3}$	+	$2 x^{2}$	+	$0 x$	+	$5$
			-	$7 x^{3}$	-	$7 x^{2}$
					-	$5 x^{2}$	+	$0 x$
					+	$5 x^{2}$	+	$5 x$
							+	$5 x$	+	$5$
							-	$5 x$	-	$5$
										$0$

Step 3.5.16

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

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

Step 3.6

Write $7 x^{3} + 2 x^{2} + 5$ as a set of factors.

$- 8 x^{2} + (x + 1) (7 x^{2} - 5 x + 5)$

Step 4

Factor $- 1$ out of $- 8 x^{2} + (x + 1) (7 x^{2} - 5 x + 5)$ .

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Step 4.1

Reorder $- 8 x^{2}$ and $(x + 1) (7 x^{2} - 5 x + 5)$ .

$(x + 1) (7 x^{2} - 5 x + 5) - 8 x^{2}$

Step 4.2

Factor $- 1$ out of $(x + 1) (7 x^{2} - 5 x + 5)$ .

$- ((- x - 1) (7 x^{2} - 5 x + 5)) - 8 x^{2}$

Step 4.3

Factor $- 1$ out of $- 8 x^{2}$ .

$- ((- x - 1) (7 x^{2} - 5 x + 5)) - (8 x^{2})$

Step 4.4

Factor $- 1$ out of $- ((- x - 1) (7 x^{2} - 5 x + 5)) - (8 x^{2})$ .

$- ((- x - 1) (7 x^{2} - 5 x + 5) + 8 x^{2})$

Step 5

Expand $(- x - 1) (7 x^{2} - 5 x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$- (- x (7 x^{2}) - x (- 5 x) - x \cdot 5 - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6

Simplify each term.

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Step 6.1

Rewrite using the commutative property of multiplication.

$- (- 1 \cdot 7 x \cdot x^{2} - x (- 5 x) - x \cdot 5 - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 6.2.1

Move $x^{2}$ .

$- (- 1 \cdot 7 (x^{2} x) - x (- 5 x) - x \cdot 5 - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.2.2

Multiply $x^{2}$ by $x$ .

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Step 6.2.2.1

Raise $x$ to the power of $1$ .

$- (- 1 \cdot 7 (x^{2} x^{1}) - x (- 5 x) - x \cdot 5 - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- (- 1 \cdot 7 x^{2 + 1} - x (- 5 x) - x \cdot 5 - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.2.3

Add $2$ and $1$ .

$- (- 1 \cdot 7 x^{3} - x (- 5 x) - x \cdot 5 - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.3

Multiply $- 1$ by $7$ .

$- (- 7 x^{3} - x (- 5 x) - x \cdot 5 - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.4

Rewrite using the commutative property of multiplication.

$- (- 7 x^{3} - 1 \cdot - 5 x \cdot x - x \cdot 5 - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.5

Multiply $x$ by $x$ by adding the exponents.

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Step 6.5.1

Move $x$ .

$- (- 7 x^{3} - 1 \cdot - 5 (x \cdot x) - x \cdot 5 - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.5.2

Multiply $x$ by $x$ .

$- (- 7 x^{3} - 1 \cdot - 5 x^{2} - x \cdot 5 - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.6

Multiply $- 1$ by $- 5$ .

$- (- 7 x^{3} + 5 x^{2} - x \cdot 5 - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.7

Multiply $5$ by $- 1$ .

$- (- 7 x^{3} + 5 x^{2} - 5 x - 1 (7 x^{2}) - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.8

Multiply $7$ by $- 1$ .

$- (- 7 x^{3} + 5 x^{2} - 5 x - 7 x^{2} - 1 (- 5 x) - 1 \cdot 5 + 8 x^{2})$

Step 6.9

Multiply $- 5$ by $- 1$ .

$- (- 7 x^{3} + 5 x^{2} - 5 x - 7 x^{2} + 5 x - 1 \cdot 5 + 8 x^{2})$

Step 6.10

Multiply $- 1$ by $5$ .

$- (- 7 x^{3} + 5 x^{2} - 5 x - 7 x^{2} + 5 x - 5 + 8 x^{2})$

Step 7

Combine the opposite terms in $- 7 x^{3} + 5 x^{2} - 5 x - 7 x^{2} + 5 x - 5$ .

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Step 7.1

Add $- 5 x$ and $5 x$ .

$- (- 7 x^{3} + 5 x^{2} - 7 x^{2} + 0 - 5 + 8 x^{2})$

Step 7.2

Add $- 7 x^{3} + 5 x^{2} - 7 x^{2}$ and $0$ .

$- (- 7 x^{3} + 5 x^{2} - 7 x^{2} - 5 + 8 x^{2})$

Step 8

Subtract $7 x^{2}$ from $5 x^{2}$ .

$- (- 7 x^{3} - 2 x^{2} - 5 + 8 x^{2})$

Step 9

Add $- 2 x^{2}$ and $8 x^{2}$ .

$- (- 7 x^{3} + 6 x^{2} - 5)$

Step 10

Factor.

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Step 10.1

Factor $- 1$ out of $- 7 x^{3} + 6 x^{2} - 5$ .

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Step 10.1.1

Factor $- 1$ out of $- 7 x^{3}$ .

$- (- (7 x^{3}) + 6 x^{2} - 5)$

Step 10.1.2

Factor $- 1$ out of $6 x^{2}$ .

$- (- (7 x^{3}) - (- 6 x^{2}) - 5)$

Step 10.1.3

Rewrite $- 5$ as $- 1 (5)$ .

$- (- (7 x^{3}) - (- 6 x^{2}) - 1 (5))$

Step 10.1.4

Factor $- 1$ out of $- (7 x^{3}) - (- 6 x^{2})$ .

$- (- (7 x^{3} - 6 x^{2}) - 1 (5))$

Step 10.1.5

Factor $- 1$ out of $- (7 x^{3} - 6 x^{2}) - 1 (5)$ .

$- - (7 x^{3} - 6 x^{2} + 5)$

Step 10.2

Remove unnecessary parentheses.

$- - 1 (7 x^{3} - 6 x^{2} + 5)$

Step 11

Combine exponents.

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Step 11.1

Factor out negative.

$- - (7 x^{3} - 6 x^{2} + 5)$

Step 11.2

Multiply $- 1$ by $- 1$ .

$1 (7 x^{3} - 6 x^{2} + 5)$

Step 11.3

Multiply $7 x^{3} - 6 x^{2} + 5$ by $1$ .

$7 x^{3} - 6 x^{2} + 5$