Calculus Examples

$y = 2 x^{3} - 3 x^{2} - 36 x + 5$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 2 x^{3} - 3 x^{2} - 36 x + 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 x^{3} - 3 x^{2} - 36 x + 5 = 0$ .

$2 x^{3} - 3 x^{2} - 36 x + 5 = 0$

Step 1.2.2

Factor $2 x^{3} - 3 x^{2} - 36 x + 5$ using the rational roots test.

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Step 1.2.2.1

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.

$p = \pm 1, \pm 5$

$q = \pm 1, \pm 2$

Step 1.2.2.2

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

$\pm 1, \pm 0.5, \pm 5, \pm 2.5$

Step 1.2.2.3

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 1.2.2.3.1

Substitute $5$ into the polynomial.

$2 \cdot 5^{3} - 3 \cdot 5^{2} - 36 \cdot 5 + 5$

Step 1.2.2.3.2

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

$2 \cdot 125 - 3 \cdot 5^{2} - 36 \cdot 5 + 5$

Step 1.2.2.3.3

Multiply $2$ by $125$ .

$250 - 3 \cdot 5^{2} - 36 \cdot 5 + 5$

Step 1.2.2.3.4

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

$250 - 3 \cdot 25 - 36 \cdot 5 + 5$

Step 1.2.2.3.5

Multiply $- 3$ by $25$ .

$250 - 75 - 36 \cdot 5 + 5$

Step 1.2.2.3.6

Subtract $75$ from $250$ .

$175 - 36 \cdot 5 + 5$

Step 1.2.2.3.7

Multiply $- 36$ by $5$ .

$175 - 180 + 5$

Step 1.2.2.3.8

Subtract $180$ from $175$ .

$- 5 + 5$

Step 1.2.2.3.9

Add $- 5$ and $5$ .

$0$

Step 1.2.2.4

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.

$\frac{2 x^{3} - 3 x^{2} - 36 x + 5}{x - 5}$

Step 1.2.2.5

Divide $2 x^{3} - 3 x^{2} - 36 x + 5$ by $x - 5$ .

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Step 1.2.2.5.1

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

$x$

$5$

$2 x^{3}$

$3 x^{2}$

$36 x$

$5$

Step 1.2.2.5.2

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

					$2 x^{2}$
	$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$

Step 1.2.2.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			+	$2 x^{3}$	-	$10 x^{2}$

Step 1.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 10 x^{2}$

				$2 x^{2}$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$

Step 1.2.2.5.5

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

				$2 x^{2}$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$7 x^{2}$

Step 1.2.2.5.6

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

				$2 x^{2}$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$7 x^{2}$	-	$36 x$

Step 1.2.2.5.7

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$7 x$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$7 x^{2}$	-	$36 x$

Step 1.2.2.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$7 x$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$7 x^{2}$	-	$36 x$
					+	$7 x^{2}$	-	$35 x$

Step 1.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} - 35 x$

				$2 x^{2}$	+	$7 x$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$7 x^{2}$	-	$36 x$
					-	$7 x^{2}$	+	$35 x$

Step 1.2.2.5.10

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

				$2 x^{2}$	+	$7 x$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$7 x^{2}$	-	$36 x$
					-	$7 x^{2}$	+	$35 x$
							-	$x$

Step 1.2.2.5.11

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

				$2 x^{2}$	+	$7 x$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$7 x^{2}$	-	$36 x$
					-	$7 x^{2}$	+	$35 x$
							-	$x$	+	$5$

Step 1.2.2.5.12

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

				$2 x^{2}$	+	$7 x$	-	$1$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$7 x^{2}$	-	$36 x$
					-	$7 x^{2}$	+	$35 x$
							-	$x$	+	$5$

Step 1.2.2.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$7 x$	-	$1$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$7 x^{2}$	-	$36 x$
					-	$7 x^{2}$	+	$35 x$
							-	$x$	+	$5$
							-	$x$	+	$5$

Step 1.2.2.5.14

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

				$2 x^{2}$	+	$7 x$	-	$1$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$7 x^{2}$	-	$36 x$
					-	$7 x^{2}$	+	$35 x$
							-	$x$	+	$5$
							+	$x$	-	$5$

Step 1.2.2.5.15

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

				$2 x^{2}$	+	$7 x$	-	$1$
$x$	-	$5$		$2 x^{3}$	-	$3 x^{2}$	-	$36 x$	+	$5$
			-	$2 x^{3}$	+	$10 x^{2}$
					+	$7 x^{2}$	-	$36 x$
					-	$7 x^{2}$	+	$35 x$
							-	$x$	+	$5$
							+	$x$	-	$5$
										$0$

Step 1.2.2.5.16

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

$2 x^{2} + 7 x - 1$

Step 1.2.2.6

Write $2 x^{3} - 3 x^{2} - 36 x + 5$ as a set of factors.

$(x - 5) (2 x^{2} + 7 x - 1) = 0$

Step 1.2.3

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

$x - 5 = 0$

$2 x^{2} + 7 x - 1 = 0$

Step 1.2.4

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 1.2.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 1.2.5

Set $2 x^{2} + 7 x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $2 x^{2} + 7 x - 1$ equal to $0$ .

$2 x^{2} + 7 x - 1 = 0$

Step 1.2.5.2

Solve $2 x^{2} + 7 x - 1 = 0$ for $x$ .

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Step 1.2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 1.2.5.2.2

Substitute the values $a = 2$ , $b = 7$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Step 1.2.5.2.3

Simplify.

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Step 1.2.5.2.3.1

Simplify the numerator.

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Step 1.2.5.2.3.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 1.2.5.2.3.1.2

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

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Step 1.2.5.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 8 \cdot - 1}}{2 \cdot 2}$

Step 1.2.5.2.3.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{- 7 \pm \sqrt{49 + 8}}{2 \cdot 2}$

Step 1.2.5.2.3.1.3

Add $49$ and $8$ .

$x = \frac{- 7 \pm \sqrt{57}}{2 \cdot 2}$

Step 1.2.5.2.3.2

Multiply $2$ by $2$ .

$x = \frac{- 7 \pm \sqrt{57}}{4}$

Step 1.2.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.5.2.4.1

Simplify the numerator.

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Step 1.2.5.2.4.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 1.2.5.2.4.1.2

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

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Step 1.2.5.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 8 \cdot - 1}}{2 \cdot 2}$

Step 1.2.5.2.4.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{- 7 \pm \sqrt{49 + 8}}{2 \cdot 2}$

Step 1.2.5.2.4.1.3

Add $49$ and $8$ .

$x = \frac{- 7 \pm \sqrt{57}}{2 \cdot 2}$

Step 1.2.5.2.4.2

Multiply $2$ by $2$ .

$x = \frac{- 7 \pm \sqrt{57}}{4}$

Step 1.2.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 7 + \sqrt{57}}{4}$

Step 1.2.5.2.4.4

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

$x = \frac{- 1 \cdot 7 + \sqrt{57}}{4}$

Step 1.2.5.2.4.5

Factor $- 1$ out of $\sqrt{57}$ .

$x = \frac{- 1 \cdot 7 - 1 (- \sqrt{57})}{4}$

Step 1.2.5.2.4.6

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{57})$ .

$x = \frac{- 1 (7 - \sqrt{57})}{4}$

Step 1.2.5.2.4.7

Move the negative in front of the fraction.

$x = - \frac{7 - \sqrt{57}}{4}$

Step 1.2.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.5.2.5.1

Simplify the numerator.

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Step 1.2.5.2.5.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 1.2.5.2.5.1.2

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

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Step 1.2.5.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 8 \cdot - 1}}{2 \cdot 2}$

Step 1.2.5.2.5.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{- 7 \pm \sqrt{49 + 8}}{2 \cdot 2}$

Step 1.2.5.2.5.1.3

Add $49$ and $8$ .

$x = \frac{- 7 \pm \sqrt{57}}{2 \cdot 2}$

Step 1.2.5.2.5.2

Multiply $2$ by $2$ .

$x = \frac{- 7 \pm \sqrt{57}}{4}$

Step 1.2.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 7 - \sqrt{57}}{4}$

Step 1.2.5.2.5.4

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

$x = \frac{- 1 \cdot 7 - \sqrt{57}}{4}$

Step 1.2.5.2.5.5

Factor $- 1$ out of $- \sqrt{57}$ .

$x = \frac{- 1 \cdot 7 - (\sqrt{57})}{4}$

Step 1.2.5.2.5.6

Factor $- 1$ out of $- 1 (7) - (\sqrt{57})$ .

$x = \frac{- 1 (7 + \sqrt{57})}{4}$

Step 1.2.5.2.5.7

Move the negative in front of the fraction.

$x = - \frac{7 + \sqrt{57}}{4}$

Step 1.2.5.2.6

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{57}}{4}, - \frac{7 + \sqrt{57}}{4}$

Step 1.2.6

The final solution is all the values that make $(x - 5) (2 x^{2} + 7 x - 1) = 0$ true.

$x = 5, - \frac{7 - \sqrt{57}}{4}, - \frac{7 + \sqrt{57}}{4}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(5, 0), (- \frac{7 - \sqrt{57}}{4}, 0), (- \frac{7 + \sqrt{57}}{4}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 2 {(0)}^{3} - 3 {(0)}^{2} - 36 \cdot 0 + 5$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \cdot 0^{3} - 3 {(0)}^{2} - 36 \cdot 0 + 5$

Step 2.2.2

Remove parentheses.

$y = 2 \cdot 0^{3} - 3 \cdot 0^{2} - 36 \cdot 0 + 5$

Step 2.2.3

Remove parentheses.

$y = 2 {(0)}^{3} - 3 {(0)}^{2} - 36 \cdot 0 + 5$

Step 2.2.4

Simplify $2 {(0)}^{3} - 3 {(0)}^{2} - 36 \cdot 0 + 5$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = 2 \cdot 0 - 3 {(0)}^{2} - 36 \cdot 0 + 5$

Step 2.2.4.1.2

Multiply $2$ by $0$ .

$y = 0 - 3 {(0)}^{2} - 36 \cdot 0 + 5$

Step 2.2.4.1.3

Raising $0$ to any positive power yields $0$ .

$y = 0 - 3 \cdot 0 - 36 \cdot 0 + 5$

Step 2.2.4.1.4

Multiply $- 3$ by $0$ .

$y = 0 + 0 - 36 \cdot 0 + 5$

Step 2.2.4.1.5

Multiply $- 36$ by $0$ .

$y = 0 + 0 + 0 + 5$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 5$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0 + 5$

Step 2.2.4.2.3

Add $0$ and $5$ .

$y = 5$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 5)$

Step 3

List the intersections.

x-intercept(s): $(5, 0), (- \frac{7 - \sqrt{57}}{4}, 0), (- \frac{7 + \sqrt{57}}{4}, 0)$

y-intercept(s): $(0, 5)$

Step 4