Calculus Examples

$f (x) = x^{3} - 9 x^{2} + 24 x - 18$

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 $x$ .

$0 = x^{3} - 9 x^{2} + 24 x - 18$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{3} - 9 x^{2} + 24 x - 18 = 0$ .

$x^{3} - 9 x^{2} + 24 x - 18 = 0$

Step 1.2.2

Factor $x^{3} - 9 x^{2} + 24 x - 18$ 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.

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

$q = \pm 1$

Step 1.2.2.2

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

$\pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6$

Step 1.2.2.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 1.2.2.3.1

Substitute $3$ into the polynomial.

$3^{3} - 9 \cdot 3^{2} + 24 \cdot 3 - 18$

Step 1.2.2.3.2

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

$27 - 9 \cdot 3^{2} + 24 \cdot 3 - 18$

Step 1.2.2.3.3

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

$27 - 9 \cdot 9 + 24 \cdot 3 - 18$

Step 1.2.2.3.4

Multiply $- 9$ by $9$ .

$27 - 81 + 24 \cdot 3 - 18$

Step 1.2.2.3.5

Subtract $81$ from $27$ .

$- 54 + 24 \cdot 3 - 18$

Step 1.2.2.3.6

Multiply $24$ by $3$ .

$- 54 + 72 - 18$

Step 1.2.2.3.7

Add $- 54$ and $72$ .

$18 - 18$

Step 1.2.2.3.8

Subtract $18$ from $18$ .

$0$

Step 1.2.2.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 9 x^{2} + 24 x - 18}{x - 3}$

Step 1.2.2.5

Divide $x^{3} - 9 x^{2} + 24 x - 18$ by $x - 3$ .

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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 $0$ .

$x$

$3$

$x^{3}$

$9 x^{2}$

$24 x$

$18$

Step 1.2.2.5.2

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

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

Step 1.2.2.5.3

Multiply the new quotient term by the divisor.

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

Step 1.2.2.5.4

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

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

Step 1.2.2.5.5

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

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

Step 1.2.2.5.6

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

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

Step 1.2.2.5.7

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

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

Step 1.2.2.5.8

Multiply the new quotient term by the divisor.

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

Step 1.2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} + 18 x$

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

Step 1.2.2.5.10

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

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

Step 1.2.2.5.11

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

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

Step 1.2.2.5.12

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $x$ .

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

Step 1.2.2.5.13

Multiply the new quotient term by the divisor.

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

Step 1.2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x - 18$

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

Step 1.2.2.5.15

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

				$x^{2}$	-	$6 x$	+	$6$
$x$	-	$3$		$x^{3}$	-	$9 x^{2}$	+	$24 x$	-	$18$
			-	$x^{3}$	+	$3 x^{2}$
					-	$6 x^{2}$	+	$24 x$
					+	$6 x^{2}$	-	$18 x$
							+	$6 x$	-	$18$
							-	$6 x$	+	$18$
										$0$

Step 1.2.2.5.16

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

$x^{2} - 6 x + 6$

Step 1.2.2.6

Write $x^{3} - 9 x^{2} + 24 x - 18$ as a set of factors.

$(x - 3) (x^{2} - 6 x + 6) = 0$

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 $0$ .

$x - 3 = 0$

$x^{2} - 6 x + 6 = 0$

Step 1.2.4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 1.2.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.5

Set $x^{2} - 6 x + 6$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x^{2} - 6 x + 6$ equal to $0$ .

$x^{2} - 6 x + 6 = 0$

Step 1.2.5.2

Solve $x^{2} - 6 x + 6 = 0$ 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 $a = 1$ , $b = - 6$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 6)}}{2 \cdot 1}$

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 $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 1.2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 1.2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 6}}{2 \cdot 1}$

Step 1.2.5.2.3.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{6 \pm \sqrt{36 - 24}}{2 \cdot 1}$

Step 1.2.5.2.3.1.3

Subtract $24$ from $36$ .

$x = \frac{6 \pm \sqrt{12}}{2 \cdot 1}$

Step 1.2.5.2.3.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.5.2.3.1.4.1

Factor $4$ out of $12$ .

$x = \frac{6 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 1.2.5.2.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 1.2.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{3}}{2}$

Step 1.2.5.2.3.3

Simplify $\frac{6 \pm 2 \sqrt{3}}{2}$ .

$x = 3 \pm \sqrt{3}$

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 $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 1.2.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 1.2.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 6}}{2 \cdot 1}$

Step 1.2.5.2.4.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{6 \pm \sqrt{36 - 24}}{2 \cdot 1}$

Step 1.2.5.2.4.1.3

Subtract $24$ from $36$ .

$x = \frac{6 \pm \sqrt{12}}{2 \cdot 1}$

Step 1.2.5.2.4.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.5.2.4.1.4.1

Factor $4$ out of $12$ .

$x = \frac{6 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 1.2.5.2.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.5.2.4.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 1.2.5.2.4.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{3}}{2}$

Step 1.2.5.2.4.3

Simplify $\frac{6 \pm 2 \sqrt{3}}{2}$ .

$x = 3 \pm \sqrt{3}$

Step 1.2.5.2.4.4

Change the $\pm$ to $+$ .

$x = 3 + \sqrt{3}$

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 $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 1.2.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 1.2.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 6}}{2 \cdot 1}$

Step 1.2.5.2.5.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{6 \pm \sqrt{36 - 24}}{2 \cdot 1}$

Step 1.2.5.2.5.1.3

Subtract $24$ from $36$ .

$x = \frac{6 \pm \sqrt{12}}{2 \cdot 1}$

Step 1.2.5.2.5.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.5.2.5.1.4.1

Factor $4$ out of $12$ .

$x = \frac{6 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 1.2.5.2.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{6 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.5.2.5.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 1.2.5.2.5.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{3}}{2}$

Step 1.2.5.2.5.3

Simplify $\frac{6 \pm 2 \sqrt{3}}{2}$ .

$x = 3 \pm \sqrt{3}$

Step 1.2.5.2.5.4

Change the $\pm$ to $-$ .

$x = 3 - \sqrt{3}$

Step 1.2.5.2.6

The final answer is the combination of both solutions.

$x = 3 + \sqrt{3}, 3 - \sqrt{3}$

Step 1.2.6

The final solution is all the values that make $(x - 3) (x^{2} - 6 x + 6) = 0$ true.

$x = 3, 3 + \sqrt{3}, 3 - \sqrt{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (3 + \sqrt{3}, 0), (3 - \sqrt{3}, 0)$

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 $y$ .

$y = {(0)}^{3} - 9 {(0)}^{2} + 24 (0) - 18$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{3} - 9 {(0)}^{2} + 24 (0) - 18$

Step 2.2.2

Remove parentheses.

$y = 0^{3} - 9 \cdot 0^{2} + 24 (0) - 18$

Step 2.2.3

Remove parentheses.

$y = {(0)}^{3} - 9 {(0)}^{2} + 24 (0) - 18$

Step 2.2.4

Simplify ${(0)}^{3} - 9 {(0)}^{2} + 24 (0) - 18$ .

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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 $0$ .

$y = 0 - 9 {(0)}^{2} + 24 (0) - 18$

Step 2.2.4.1.2

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

$y = 0 - 9 \cdot 0 + 24 (0) - 18$

Step 2.2.4.1.3

Multiply $- 9$ by $0$ .

$y = 0 + 0 + 24 (0) - 18$

Step 2.2.4.1.4

Multiply $24$ by $0$ .

$y = 0 + 0 + 0 - 18$

Step 2.2.4.2

Simplify by adding and subtracting.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 - 18$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0 - 18$

Step 2.2.4.2.3

Subtract $18$ from $0$ .

$y = - 18$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 18)$

Step 3

List the intersections.

x-intercept(s): $(3, 0), (3 + \sqrt{3}, 0), (3 - \sqrt{3}, 0)$

y-intercept(s): $(0, - 18)$

Step 4