Calculus Examples

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

Step 1

Write $x^{4} - 4 x^{3}$ as an equation.

$y = x^{4} - 4 x^{3}$

Step 2

Find the x-intercepts.

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Step 2.1

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

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

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $x^{4} - 4 x^{3} = 0$ .

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

Step 2.2.2

Factor $x^{3}$ out of $x^{4} - 4 x^{3}$ .

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Step 2.2.2.1

Factor $x^{3}$ out of $x^{4}$ .

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

Step 2.2.2.2

Factor $x^{3}$ out of $- 4 x^{3}$ .

$x^{3} x + x^{3} \cdot - 4 = 0$

Step 2.2.2.3

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot - 4$ .

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

Step 2.2.3

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

$x^{3} = 0$

$x - 4 = 0$

Step 2.2.4

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

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Step 2.2.4.1

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

$x^{3} = 0$

Step 2.2.4.2

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

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Step 2.2.4.2.1

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

$x = \sqrt[3]{0}$

Step 2.2.4.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.2.4.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 2.2.4.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 2.2.5

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.2.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.2.6

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

$x = 0, 4$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (4, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = {(0)}^{4} - 4 {(0)}^{3}$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = 0^{4} - 4 {(0)}^{3}$

Step 3.2.2

Remove parentheses.

$y = 0^{4} - 4 \cdot 0^{3}$

Step 3.2.3

Remove parentheses.

$y = {(0)}^{4} - 4 {(0)}^{3}$

Step 3.2.4

Simplify ${(0)}^{4} - 4 {(0)}^{3}$ .

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

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

$y = 0 - 4 {(0)}^{3}$

Step 3.2.4.1.2

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

$y = 0 - 4 \cdot 0$

Step 3.2.4.1.3

Multiply $- 4$ by $0$ .

$y = 0 + 0$

Step 3.2.4.2

Add $0$ and $0$ .

$y = 0$

Step 3.3

y-intercept(s) in point form.

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

Step 4

List the intersections.

x-intercept(s): $(0, 0), (4, 0)$

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

Step 5