Calculus Examples

$y = x^{9} - 9 x$

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 $x$ .

$0 = x^{9} - 9 x$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{9} - 9 x = 0$ .

$x^{9} - 9 x = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $x$ out of $x^{9} - 9 x$ .

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Step 1.2.2.1.1

Factor $x$ out of $x^{9}$ .

$x \cdot x^{8} - 9 x = 0$

Step 1.2.2.1.2

Factor $x$ out of $- 9 x$ .

$x \cdot x^{8} + x \cdot - 9 = 0$

Step 1.2.2.1.3

Factor $x$ out of $x \cdot x^{8} + x \cdot - 9$ .

$x (x^{8} - 9) = 0$

Step 1.2.2.2

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

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

Step 1.2.2.3

Rewrite $9$ as $3^{2}$ .

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

Step 1.2.2.4

Factor.

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Step 1.2.2.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{4}$ and $b = 3$ .

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

Step 1.2.2.4.2

Remove unnecessary parentheses.

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

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 $0$ .

$x = 0$

$x^{4} + 3 = 0$

$x^{4} - 3 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

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

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Step 1.2.5.1

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

$x^{4} + 3 = 0$

Step 1.2.5.2

Solve $x^{4} + 3 = 0$ for $x$ .

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Step 1.2.5.2.1

Subtract $3$ from both sides of the equation.

$x^{4} = - 3$

Step 1.2.5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{- 3}$

Step 1.2.5.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.5.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt[4]{- 3}$

Step 1.2.5.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt[4]{- 3}$

Step 1.2.5.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt[4]{- 3}, - \sqrt[4]{- 3}$

Step 1.2.6

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

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Step 1.2.6.1

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

$x^{4} - 3 = 0$

Step 1.2.6.2

Solve $x^{4} - 3 = 0$ for $x$ .

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Step 1.2.6.2.1

Add $3$ to both sides of the equation.

$x^{4} = 3$

Step 1.2.6.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{3}$

Step 1.2.6.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.6.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt[4]{3}$

Step 1.2.6.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt[4]{3}$

Step 1.2.6.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt[4]{3}, - \sqrt[4]{3}$

Step 1.2.7

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

$x = 0, \sqrt[4]{- 3}, - \sqrt[4]{- 3}, \sqrt[4]{3}, - \sqrt[4]{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\sqrt[4]{- 3}, 0), (- \sqrt[4]{- 3}, 0), (\sqrt[4]{3}, 0), (- \sqrt[4]{3}, 0)$

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 $y$ .

$y = {(0)}^{9} - 9 \cdot 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{9} - 9 \cdot 0$

Step 2.2.2

Remove parentheses.

$y = {(0)}^{9} - 9 \cdot 0$

Step 2.2.3

Simplify ${(0)}^{9} - 9 \cdot 0$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

$y = 0 - 9 \cdot 0$

Step 2.2.3.1.2

Multiply $- 9$ by $0$ .

$y = 0 + 0$

Step 2.2.3.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\sqrt[4]{- 3}, 0), (- \sqrt[4]{- 3}, 0), (\sqrt[4]{3}, 0), (- \sqrt[4]{3}, 0)$

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

Step 4