Calculus Examples

$f (x) = 2 x^{3} - x^{2} - 9 x - 4$

Step 1

Set $2 x^{3} - x^{2} - 9 x - 4$ equal to $0$ .

$2 x^{3} - x^{2} - 9 x - 4 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $2 x^{3} - x^{2} - 9 x - 4$ 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.

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

$q = \pm 1, \pm 2$

Step 2.1.2

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

$\pm 1, \pm 0.5, \pm 4, \pm 2$

Step 2.1.3

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

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Step 2.1.3.1

Substitute $- 0.5$ into the polynomial.

$2 {(- 0.5)}^{3} - {(- 0.5)}^{2} - 9 \cdot - 0.5 - 4$

Step 2.1.3.2

Raise $- 0.5$ to the power of $3$ .

$2 \cdot - 0.125 - {(- 0.5)}^{2} - 9 \cdot - 0.5 - 4$

Step 2.1.3.3

Multiply $2$ by $- 0.125$ .

$- 0.25 - {(- 0.5)}^{2} - 9 \cdot - 0.5 - 4$

Step 2.1.3.4

Raise $- 0.5$ to the power of $2$ .

$- 0.25 - 1 \cdot 0.25 - 9 \cdot - 0.5 - 4$

Step 2.1.3.5

Multiply $- 1$ by $0.25$ .

$- 0.25 - 0.25 - 9 \cdot - 0.5 - 4$

Step 2.1.3.6

Subtract $0.25$ from $- 0.25$ .

$- 0.5 - 9 \cdot - 0.5 - 4$

Step 2.1.3.7

Multiply $- 9$ by $- 0.5$ .

$- 0.5 + 4.5 - 4$

Step 2.1.3.8

Add $- 0.5$ and $4.5$ .

$4 - 4$

Step 2.1.3.9

Subtract $4$ from $4$ .

$0$

Step 2.1.4

Since $- 0.5$ is a known root, divide the polynomial by $2 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - x^{2} - 9 x - 4}{2 x + 1}$

Step 2.1.5

Divide $2 x^{3} - x^{2} - 9 x - 4$ by $2 x + 1$ .

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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 $0$ .

$2 x$

$1$

$2 x^{3}$

$x^{2}$

$9 x$

$4$

Step 2.1.5.2

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

					$x^{2}$
	$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			+	$2 x^{3}$	+	$x^{2}$

Step 2.1.5.4

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

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$

Step 2.1.5.5

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

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$2 x^{2}$

Step 2.1.5.6

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

				$x^{2}$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	-	$9 x$

Step 2.1.5.7

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

				$x^{2}$	-	$x$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	-	$9 x$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	-	$9 x$
					-	$2 x^{2}$	-	$x$

Step 2.1.5.9

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

				$x^{2}$	-	$x$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	-	$9 x$
					+	$2 x^{2}$	+	$x$

Step 2.1.5.10

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

				$x^{2}$	-	$x$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	-	$9 x$
					+	$2 x^{2}$	+	$x$
							-	$8 x$

Step 2.1.5.11

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

				$x^{2}$	-	$x$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	-	$9 x$
					+	$2 x^{2}$	+	$x$
							-	$8 x$	-	$4$

Step 2.1.5.12

Divide the highest order term in the dividend $- 8 x$ by the highest order term in divisor $2 x$ .

				$x^{2}$	-	$x$	-	$4$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	-	$9 x$
					+	$2 x^{2}$	+	$x$
							-	$8 x$	-	$4$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$4$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	-	$9 x$
					+	$2 x^{2}$	+	$x$
							-	$8 x$	-	$4$
							-	$8 x$	-	$4$

Step 2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x - 4$

				$x^{2}$	-	$x$	-	$4$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	-	$9 x$
					+	$2 x^{2}$	+	$x$
							-	$8 x$	-	$4$
							+	$8 x$	+	$4$

Step 2.1.5.15

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

				$x^{2}$	-	$x$	-	$4$
$2 x$	+	$1$		$2 x^{3}$	-	$x^{2}$	-	$9 x$	-	$4$
			-	$2 x^{3}$	-	$x^{2}$
					-	$2 x^{2}$	-	$9 x$
					+	$2 x^{2}$	+	$x$
							-	$8 x$	-	$4$
							+	$8 x$	+	$4$
										$0$

Step 2.1.5.16

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

$x^{2} - x - 4$

Step 2.1.6

Write $2 x^{3} - x^{2} - 9 x - 4$ as a set of factors.

$(2 x + 1) (x^{2} - x - 4) = 0$

Step 2.2

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

$2 x + 1 = 0$

$x^{2} - x - 4 = 0$

Step 2.3

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

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Step 2.3.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 2.3.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 2.3.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 2.3.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 2.3.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 2.4

Set $x^{2} - x - 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2} - x - 4$ equal to $0$ .

$x^{2} - x - 4 = 0$

Step 2.4.2

Solve $x^{2} - x - 4 = 0$ for $x$ .

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Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 1$ , $b = - 1$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 2.4.2.3

Simplify.

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Step 2.4.2.3.1

Simplify the numerator.

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Step 2.4.2.3.1.1

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

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

Step 2.4.2.3.1.2

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

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.4.2.3.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{1 \pm \sqrt{1 + 16}}{2 \cdot 1}$

Step 2.4.2.3.1.3

Add $1$ and $16$ .

$x = \frac{1 \pm \sqrt{17}}{2 \cdot 1}$

Step 2.4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{17}}{2}$

Step 2.4.2.4

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{17}}{2}, \frac{1 - \sqrt{17}}{2}$

Step 2.5

The final solution is all the values that make $(2 x + 1) (x^{2} - x - 4) = 0$ true.

$x = - \frac{1}{2}, \frac{1 + \sqrt{17}}{2}, \frac{1 - \sqrt{17}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}, \frac{1 + \sqrt{17}}{2}, \frac{1 - \sqrt{17}}{2}$

Decimal Form:

$x = - 0.5, 2.56155281 \dots, - 1.56155281 \dots$

Step 4