Calculus Examples

$\ln (x^{2} + 1) - 3 \ln (x) = \ln (2)$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\ln (x^{2} + 1) - 3 \ln (x) - \ln (2) = 0$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{x^{2} + 1}{2}) - 3 \ln (x) = 0$

Step 3

Simplify the left side.

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Step 3.1

Simplify $\ln (\frac{x^{2} + 1}{2}) - 3 \ln (x)$ .

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Step 3.1.1

Simplify $- 3 \ln (x)$ by moving $3$ inside the logarithm.

$\ln (\frac{x^{2} + 1}{2}) - \ln (x^{3}) = 0$

Step 3.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{\frac{x^{2} + 1}{2}}{x^{3}}) = 0$

Step 3.1.3

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{x^{2} + 1}{2} \cdot \frac{1}{x^{3}}) = 0$

Step 3.1.4

Multiply $\frac{x^{2} + 1}{2}$ by $\frac{1}{x^{3}}$ .

$\ln (\frac{x^{2} + 1}{2 x^{3}}) = 0$

Step 4

Rewrite $\ln (\frac{x^{2} + 1}{2 x^{3}}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = \frac{x^{2} + 1}{2 x^{3}}$

Step 5

Cross multiply to remove the fraction.

$x^{2} + 1 = e^{0} (2 x^{3})$

Step 6

Simplify $e^{0} (2 x^{3})$ .

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Step 6.1

Anything raised to $0$ is $1$ .

$x^{2} + 1 = 1 (2 x^{3})$

Step 6.2

Multiply $2 x^{3}$ by $1$ .

$x^{2} + 1 = 2 x^{3}$

Step 7

Subtract $2 x^{3}$ from both sides of the equation.

$x^{2} + 1 - 2 x^{3} = 0$

Step 8

Factor the left side of the equation.

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Step 8.1

Reorder terms.

$- 2 x^{3} + x^{2} + 1 = 0$

Step 8.2

Factor $- 2 x^{3} + x^{2} + 1$ using the rational roots test.

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Step 8.2.1

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.

$p = \pm 1$

$q = \pm 1, \pm 2$

Step 8.2.2

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

$\pm 1, \pm 0.5$

Step 8.2.3

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 8.2.3.1

Substitute $1$ into the polynomial.

$- 2 \cdot 1^{3} + 1^{2} + 1$

Step 8.2.3.2

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

$- 2 \cdot 1 + 1^{2} + 1$

Step 8.2.3.3

Multiply $- 2$ by $1$ .

$- 2 + 1^{2} + 1$

Step 8.2.3.4

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

$- 2 + 1 + 1$

Step 8.2.3.5

Add $- 2$ and $1$ .

$- 1 + 1$

Step 8.2.3.6

Add $- 1$ and $1$ .

$0$

Step 8.2.4

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.

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

Step 8.2.5

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

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Step 8.2.5.1

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

$x$

$1$

$2 x^{3}$

$x^{2}$

$0 x$

$1$

Step 8.2.5.2

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

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

Step 8.2.5.3

Multiply the new quotient term by the divisor.

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

Step 8.2.5.4

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

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

Step 8.2.5.5

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

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

Step 8.2.5.6

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

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

Step 8.2.5.7

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

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

Step 8.2.5.8

Multiply the new quotient term by the divisor.

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

Step 8.2.5.9

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

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

Step 8.2.5.10

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

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

Step 8.2.5.11

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

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

Step 8.2.5.12

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

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

Step 8.2.5.13

Multiply the new quotient term by the divisor.

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

Step 8.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- x + 1$

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

Step 8.2.5.15

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

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

Step 8.2.5.16

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

$- 2 x^{2} - x - 1$

Step 8.2.6

Write $- 2 x^{3} + x^{2} + 1$ as a set of factors.

$(x - 1) (- 2 x^{2} - x - 1) = 0$

Step 9

Simplify $(x - 1) (- 2 x^{2} - x - 1)$ .

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Step 9.1

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

$x (- 2 x^{2}) + x (- x) + x \cdot - 1 - 1 (- 2 x^{2}) - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2

Simplify terms.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Rewrite using the commutative property of multiplication.

$- 2 x \cdot x^{2} + x (- x) + x \cdot - 1 - 1 (- 2 x^{2}) - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2.1.2

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

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Step 9.2.1.2.1

Move $x^{2}$ .

$- 2 (x^{2} x) + x (- x) + x \cdot - 1 - 1 (- 2 x^{2}) - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2.1.2.2

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

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Step 9.2.1.2.2.1

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

$- 2 (x^{2} x^{1}) + x (- x) + x \cdot - 1 - 1 (- 2 x^{2}) - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2.1.2.2.2

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

$- 2 x^{2 + 1} + x (- x) + x \cdot - 1 - 1 (- 2 x^{2}) - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2.1.2.3

Add $2$ and $1$ .

$- 2 x^{3} + x (- x) + x \cdot - 1 - 1 (- 2 x^{2}) - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2.1.3

Rewrite using the commutative property of multiplication.

$- 2 x^{3} - x \cdot x + x \cdot - 1 - 1 (- 2 x^{2}) - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2.1.4

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

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Step 9.2.1.4.1

Move $x$ .

$- 2 x^{3} - (x \cdot x) + x \cdot - 1 - 1 (- 2 x^{2}) - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2.1.4.2

Multiply $x$ by $x$ .

$- 2 x^{3} - x^{2} + x \cdot - 1 - 1 (- 2 x^{2}) - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2.1.5

Move $- 1$ to the left of $x$ .

$- 2 x^{3} - x^{2} - 1 \cdot x - 1 (- 2 x^{2}) - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2.1.6

Rewrite $- 1 x$ as $- x$ .

$- 2 x^{3} - x^{2} - x - 1 (- 2 x^{2}) - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2.1.7

Multiply $- 2$ by $- 1$ .

$- 2 x^{3} - x^{2} - x + 2 x^{2} - 1 (- x) - 1 \cdot - 1 = 0$

Step 9.2.1.8

Multiply $- 1 (- x)$ .

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Step 9.2.1.8.1

Multiply $- 1$ by $- 1$ .

$- 2 x^{3} - x^{2} - x + 2 x^{2} + 1 x - 1 \cdot - 1 = 0$

Step 9.2.1.8.2

Multiply $x$ by $1$ .

$- 2 x^{3} - x^{2} - x + 2 x^{2} + x - 1 \cdot - 1 = 0$

Step 9.2.1.9

Multiply $- 1$ by $- 1$ .

$- 2 x^{3} - x^{2} - x + 2 x^{2} + x + 1 = 0$

Step 9.2.2

Simplify by adding terms.

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Step 9.2.2.1

Combine the opposite terms in $- 2 x^{3} - x^{2} - x + 2 x^{2} + x + 1$ .

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Step 9.2.2.1.1

Add $- x$ and $x$ .

$- 2 x^{3} - x^{2} + 2 x^{2} + 0 + 1 = 0$

Step 9.2.2.1.2

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

$- 2 x^{3} - x^{2} + 2 x^{2} + 1 = 0$

Step 9.2.2.2

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

$- 2 x^{3} + x^{2} + 1 = 0$

Step 10

Factor the left side of the equation.

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Step 10.1

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

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Step 10.1.1

Factor $- 1$ out of $- 2 x^{3}$ .

$- (2 x^{3}) + x^{2} + 1 = 0$

Step 10.1.2

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

$- (2 x^{3}) - 1 (- x^{2}) + 1 = 0$

Step 10.1.3

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

$- (2 x^{3}) - 1 (- x^{2}) - 1 \cdot - 1 = 0$

Step 10.1.4

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

$- (2 x^{3} - x^{2}) - 1 \cdot - 1 = 0$

Step 10.1.5

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

$- (2 x^{3} - x^{2} - 1) = 0$

Step 10.2

Factor.

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Step 10.2.1

Factor $2 x^{3} - x^{2} - 1$ using the rational roots test.

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Step 10.2.1.1

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.

$p = \pm 1$

$q = \pm 1, \pm 2$

Step 10.2.1.2

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

$\pm 1, \pm 0.5$

Step 10.2.1.3

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 10.2.1.3.1

Substitute $1$ into the polynomial.

$2 \cdot 1^{3} - 1^{2} - 1$

Step 10.2.1.3.2

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

$2 \cdot 1 - 1^{2} - 1$

Step 10.2.1.3.3

Multiply $2$ by $1$ .

$2 - 1^{2} - 1$

Step 10.2.1.3.4

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

$2 - 1 \cdot 1 - 1$

Step 10.2.1.3.5

Multiply $- 1$ by $1$ .

$2 - 1 - 1$

Step 10.2.1.3.6

Subtract $1$ from $2$ .

$1 - 1$

Step 10.2.1.3.7

Subtract $1$ from $1$ .

$0$

Step 10.2.1.4

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.

$\frac{2 x^{3} - x^{2} - 1}{x - 1}$

Step 10.2.1.5

Divide $2 x^{3} - x^{2} - 1$ by $x - 1$ .

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Step 10.2.1.5.1

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

$x$

$1$

$2 x^{3}$

$x^{2}$

$0 x$

$1$

Step 10.2.1.5.2

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

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

Step 10.2.1.5.3

Multiply the new quotient term by the divisor.

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

Step 10.2.1.5.4

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

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

Step 10.2.1.5.5

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

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

Step 10.2.1.5.6

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

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

Step 10.2.1.5.7

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

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

Step 10.2.1.5.8

Multiply the new quotient term by the divisor.

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

Step 10.2.1.5.9

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

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

Step 10.2.1.5.10

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

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

Step 10.2.1.5.11

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

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

Step 10.2.1.5.12

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

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

Step 10.2.1.5.13

Multiply the new quotient term by the divisor.

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

Step 10.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $x - 1$

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

Step 10.2.1.5.15

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

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

Step 10.2.1.5.16

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

$2 x^{2} + x + 1$

Step 10.2.1.6

Write $2 x^{3} - x^{2} - 1$ as a set of factors.

$- ((x - 1) (2 x^{2} + x + 1)) = 0$

Step 10.2.2

Remove unnecessary parentheses.

$- (x - 1) (2 x^{2} + x + 1) = 0$

Step 11

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$2 x^{2} + x + 1 = 0$

Step 12

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 12.2

Add $1$ to both sides of the equation.

$x = 1$

Step 13

Set $2 x^{2} + x + 1$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $2 x^{2} + x + 1$ equal to $0$ .

$2 x^{2} + x + 1 = 0$

Step 13.2

Solve $2 x^{2} + x + 1 = 0$ for $x$ .

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Step 13.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 13.2.2

Substitute the values $a = 2$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (2 \cdot 1)}}{2 \cdot 2}$

Step 13.2.3

Simplify.

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Step 13.2.3.1

Simplify the numerator.

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Step 13.2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 13.2.3.1.2

Multiply $- 4 \cdot 2 \cdot 1$ .

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Step 13.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 8 \cdot 1}}{2 \cdot 2}$

Step 13.2.3.1.2.2

Multiply $- 8$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 8}}{2 \cdot 2}$

Step 13.2.3.1.3

Subtract $8$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 7}}{2 \cdot 2}$

Step 13.2.3.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 2}$

Step 13.2.3.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 2}$

Step 13.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{7}}{2 \cdot 2}$

Step 13.2.3.2

Multiply $2$ by $2$ .

$x = \frac{- 1 \pm i \sqrt{7}}{4}$

Step 13.2.4

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{7}}{4}, - \frac{1 + i \sqrt{7}}{4}$

Step 14

The final solution is all the values that make $- (x - 1) (2 x^{2} + x + 1) = 0$ true.

$x = 1, - \frac{1 - i \sqrt{7}}{4}, - \frac{1 + i \sqrt{7}}{4}$