Calculus Examples

$f (x) = 4 x^{3} - 24 x^{2} + 35 x - 36$

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 $x$ .

$0 = 4 x^{3} - 24 x^{2} + 35 x - 36$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $4 x^{3} - 24 x^{2} + 35 x - 36 = 0$ .

$4 x^{3} - 24 x^{2} + 35 x - 36 = 0$

Step 1.2.2

Factor $4 x^{3} - 24 x^{2} + 35 x - 36$ 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.

$p = \pm 1, \pm 36, \pm 2, \pm 18, \pm 3, \pm 12, \pm 4, \pm 9, \pm 6$

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

Step 1.2.2.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 36, \pm 9, \pm 18, \pm 2, \pm 4.5, \pm 3, \pm 0.75, \pm 1.5, \pm 12, \pm 6, \pm 4, \pm 2.25$

Step 1.2.2.3

Substitute $4.5$ and simplify the expression. In this case, the expression is equal to $0$ so $4.5$ is a root of the polynomial.

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Step 1.2.2.3.1

Substitute $4.5$ into the polynomial.

$4 \cdot {4.5}^{3} - 24 \cdot {4.5}^{2} + 35 \cdot 4.5 - 36$

Step 1.2.2.3.2

Raise $4.5$ to the power of $3$ .

$4 \cdot 91.125 - 24 \cdot {4.5}^{2} + 35 \cdot 4.5 - 36$

Step 1.2.2.3.3

Multiply $4$ by $91.125$ .

$364.5 - 24 \cdot {4.5}^{2} + 35 \cdot 4.5 - 36$

Step 1.2.2.3.4

Raise $4.5$ to the power of $2$ .

$364.5 - 24 \cdot 20.25 + 35 \cdot 4.5 - 36$

Step 1.2.2.3.5

Multiply $- 24$ by $20.25$ .

$364.5 - 486 + 35 \cdot 4.5 - 36$

Step 1.2.2.3.6

Subtract $486$ from $364.5$ .

$- 121.5 + 35 \cdot 4.5 - 36$

Step 1.2.2.3.7

Multiply $35$ by $4.5$ .

$- 121.5 + 157.5 - 36$

Step 1.2.2.3.8

Add $- 121.5$ and $157.5$ .

$36 - 36$

Step 1.2.2.3.9

Subtract $36$ from $36$ .

$0$

Step 1.2.2.4

Since $4.5$ is a known root, divide the polynomial by $2 x - 9$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 24 x^{2} + 35 x - 36}{2 x - 9}$

Step 1.2.2.5

Divide $4 x^{3} - 24 x^{2} + 35 x - 36$ by $2 x - 9$ .

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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 $0$ .

$2 x$

$9$

$4 x^{3}$

$24 x^{2}$

$35 x$

$36$

Step 1.2.2.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$

Step 1.2.2.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			+	$4 x^{3}$	-	$18 x^{2}$

Step 1.2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 18 x^{2}$

				$2 x^{2}$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$

Step 1.2.2.5.5

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

				$2 x^{2}$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$
					-	$6 x^{2}$

Step 1.2.2.5.6

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

				$2 x^{2}$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$
					-	$6 x^{2}$	+	$35 x$

Step 1.2.2.5.7

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

				$2 x^{2}$	-	$3 x$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$
					-	$6 x^{2}$	+	$35 x$

Step 1.2.2.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$
					-	$6 x^{2}$	+	$35 x$
					-	$6 x^{2}$	+	$27 x$

Step 1.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} + 27 x$

				$2 x^{2}$	-	$3 x$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$
					-	$6 x^{2}$	+	$35 x$
					+	$6 x^{2}$	-	$27 x$

Step 1.2.2.5.10

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

				$2 x^{2}$	-	$3 x$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$
					-	$6 x^{2}$	+	$35 x$
					+	$6 x^{2}$	-	$27 x$
							+	$8 x$

Step 1.2.2.5.11

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

				$2 x^{2}$	-	$3 x$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$
					-	$6 x^{2}$	+	$35 x$
					+	$6 x^{2}$	-	$27 x$
							+	$8 x$	-	$36$

Step 1.2.2.5.12

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

				$2 x^{2}$	-	$3 x$	+	$4$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$
					-	$6 x^{2}$	+	$35 x$
					+	$6 x^{2}$	-	$27 x$
							+	$8 x$	-	$36$

Step 1.2.2.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$	+	$4$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$
					-	$6 x^{2}$	+	$35 x$
					+	$6 x^{2}$	-	$27 x$
							+	$8 x$	-	$36$
							+	$8 x$	-	$36$

Step 1.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $8 x - 36$

				$2 x^{2}$	-	$3 x$	+	$4$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$
					-	$6 x^{2}$	+	$35 x$
					+	$6 x^{2}$	-	$27 x$
							+	$8 x$	-	$36$
							-	$8 x$	+	$36$

Step 1.2.2.5.15

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

				$2 x^{2}$	-	$3 x$	+	$4$
$2 x$	-	$9$		$4 x^{3}$	-	$24 x^{2}$	+	$35 x$	-	$36$
			-	$4 x^{3}$	+	$18 x^{2}$
					-	$6 x^{2}$	+	$35 x$
					+	$6 x^{2}$	-	$27 x$
							+	$8 x$	-	$36$
							-	$8 x$	+	$36$
										$0$

Step 1.2.2.5.16

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

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

Step 1.2.2.6

Write $4 x^{3} - 24 x^{2} + 35 x - 36$ as a set of factors.

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

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 $0$ .

$2 x - 9 = 0$

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

Step 1.2.4

Set $2 x - 9$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $2 x - 9$ equal to $0$ .

$2 x - 9 = 0$

Step 1.2.4.2

Solve $2 x - 9 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $9$ to both sides of the equation.

$2 x = 9$

Step 1.2.4.2.2

Divide each term in $2 x = 9$ by $2$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $2 x = 9$ by $2$ .

$\frac{2 x}{2} = \frac{9}{2}$

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{9}{2}$

Step 1.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{2}$

Step 1.2.5

Set $2 x^{2} - 3 x + 4$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

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

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

Step 1.2.5.2

Solve $2 x^{2} - 3 x + 4 = 0$ 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 $a = 2$ , $b = - 3$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (2 \cdot 4)}}{2 \cdot 2}$

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 $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot 4}}{2 \cdot 2}$

Step 1.2.5.2.3.1.2

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

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Step 1.2.5.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot 4}}{2 \cdot 2}$

Step 1.2.5.2.3.1.2.2

Multiply $- 8$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 32}}{2 \cdot 2}$

Step 1.2.5.2.3.1.3

Subtract $32$ from $9$ .

$x = \frac{3 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 1.2.5.2.3.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 1.2.5.2.3.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 1.2.5.2.3.1.6

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

$x = \frac{3 \pm i \sqrt{23}}{2 \cdot 2}$

Step 1.2.5.2.3.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm i \sqrt{23}}{4}$

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 $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot 4}}{2 \cdot 2}$

Step 1.2.5.2.4.1.2

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

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Step 1.2.5.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot 4}}{2 \cdot 2}$

Step 1.2.5.2.4.1.2.2

Multiply $- 8$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 32}}{2 \cdot 2}$

Step 1.2.5.2.4.1.3

Subtract $32$ from $9$ .

$x = \frac{3 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 1.2.5.2.4.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 1.2.5.2.4.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 1.2.5.2.4.1.6

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

$x = \frac{3 \pm i \sqrt{23}}{2 \cdot 2}$

Step 1.2.5.2.4.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm i \sqrt{23}}{4}$

Step 1.2.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + i \sqrt{23}}{4}$

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 $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot 4}}{2 \cdot 2}$

Step 1.2.5.2.5.1.2

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

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Step 1.2.5.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot 4}}{2 \cdot 2}$

Step 1.2.5.2.5.1.2.2

Multiply $- 8$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 32}}{2 \cdot 2}$

Step 1.2.5.2.5.1.3

Subtract $32$ from $9$ .

$x = \frac{3 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 1.2.5.2.5.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 1.2.5.2.5.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 1.2.5.2.5.1.6

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

$x = \frac{3 \pm i \sqrt{23}}{2 \cdot 2}$

Step 1.2.5.2.5.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm i \sqrt{23}}{4}$

Step 1.2.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - i \sqrt{23}}{4}$

Step 1.2.5.2.6

The final answer is the combination of both solutions.

$x = \frac{3 + i \sqrt{23}}{4}, \frac{3 - i \sqrt{23}}{4}$

Step 1.2.6

The final solution is all the values that make $(2 x - 9) (2 x^{2} - 3 x + 4) = 0$ true.

$x = \frac{9}{2}, \frac{3 + i \sqrt{23}}{4}, \frac{3 - i \sqrt{23}}{4}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{9}{2}, 0)$

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 $y$ .

$y = 4 {(0)}^{3} - 24 {(0)}^{2} + 35 (0) - 36$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 \cdot 0^{3} - 24 {(0)}^{2} + 35 (0) - 36$

Step 2.2.2

Remove parentheses.

$y = 4 \cdot 0^{3} - 24 \cdot 0^{2} + 35 (0) - 36$

Step 2.2.3

Remove parentheses.

$y = 4 {(0)}^{3} - 24 {(0)}^{2} + 35 (0) - 36$

Step 2.2.4

Simplify $4 {(0)}^{3} - 24 {(0)}^{2} + 35 (0) - 36$ .

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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 $0$ .

$y = 4 \cdot 0 - 24 {(0)}^{2} + 35 (0) - 36$

Step 2.2.4.1.2

Multiply $4$ by $0$ .

$y = 0 - 24 {(0)}^{2} + 35 (0) - 36$

Step 2.2.4.1.3

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

$y = 0 - 24 \cdot 0 + 35 (0) - 36$

Step 2.2.4.1.4

Multiply $- 24$ by $0$ .

$y = 0 + 0 + 35 (0) - 36$

Step 2.2.4.1.5

Multiply $35$ by $0$ .

$y = 0 + 0 + 0 - 36$

Step 2.2.4.2

Simplify by adding and subtracting.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 - 36$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0 - 36$

Step 2.2.4.2.3

Subtract $36$ from $0$ .

$y = - 36$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 36)$

Step 3

List the intersections.

x-intercept(s): $(\frac{9}{2}, 0)$

y-intercept(s): $(0, - 36)$

Step 4